Bài tập ôn tập môn Toán Lớp 12 - Chuyên đề 41: Tính tích phân hàm hợp - Năm học 2021-2022 (Kèm đáp án)
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25/04/2025
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- Chuyờn đề ㊶ TÍNH TÍCH PHÂN HÀM HỢP Ⓐ KIẾN THỨC CƠ BẢN CẦN NẮM ❖ Ghi nhớ Định nghĩa • Cho hàm số f liờn tục trờn K và a, b là hai số bất kỡ thuộc K. • Nếu F là một nguyờn hàm của f trờn K thỡ hiệu số F(b)- F(a)được gọi là tớch phõn của f từ a đến b b • Kớ hiệu là : ũ f (x)dx. a b ➢ Trong trường hợp a < b, ta gọi ũ f (x)dx là tớch phõn của f trờn đoạn [a;b]. a b • Người ta cũn dựng kớ hiệu F (x) để chỉ hiệu số F (b)- F (a). a b b ➢Như vậy ta cú: f (x)dx = F (x) . ũ a a ❖ Ghi nhớ ❷ ❖ Định lý: Giả sử cỏc hàm số f , g liờn tục trờn K và a, b, c là ba số bất kỡ thuộc K. Khi đú ta cú a b a ➀. ũ f (x)dx = 0; ➁. ũ f (x)dx = - ũ f (x)dx; a a b b c c b b b ộ ự ➂. ũ f (x)dx + ũ f (x)dx = ũ f (x)dx; ➃. ũ ởf (x)+ g(x)ỷdx = ũ f (x)dx + ũ g(x)dx; a b a a a a b b ➄. ũ kf (x)dx = kũ f (x)dx với k ẻ Ă . a a ❖ Ghi nhớ ❸ ❖ Phương phỏp đổi biến số: ❖ b ộ ự Â ➢ Để tớnh tớch phõn I = ũ f (x)dx nếu f (x)= g ởu(x)ỷ.u (x), ta cú thể thực hiện phộp a đổi biến như sau: • Bước 1. Đặt t = u(x)ị dt = uÂ(x)dx. ỡ ù x = a ị t = u(a) • Bước 2. Đổi cận: ớ . ù ợù x = b ị t = u(b) u(b) u(b) • Bước 3. Thay vào, ta cú I = ũ g(t)dt = G(t) . u(a) u(a) ❖ Dấu hiệu nhận biết và cỏch đổi biến Dấu hiệu Cú thể đặt Vớ dụ
- 3 3 x dx ① Cú f (x) t f (x) I . ☞Đặt t x 1 0 x 1 1 ② Cú (ax b)n t ax b I x(x 1)2022 dx . ☞Đặt t x 1 0 etan x 3 ③ Cú a f ( x) t f (x) I 4 dx . ☞Đặt t tan x 3 0 cos2 x dx t ln x hoặc biểu thức e ln xdx ④ Cú và ln x I . ☞Đặt t ln x 1 x chứa ln x 1 x(ln x 1) x ln2 t e hoặc biểu thức chứa 2x x x ⑤ Cú exdx I e 3e 1dx .☞ Đặt t 3e 1 ex 0 ⑥ Cú sin xdx t cos x I 2 sin3 xcos xdx . ☞Đặt t sin x 0 3 sin x ⑦ Cú cos xdx t sin xdx I dx . ☞Đặt t 2 cos x 1 0 2cos x 1 1 1 dx I 4 dx 4 (1 tan2 x) dx ⑧ Cú t tan x 0 cos4 x 0 cos2 x cos2 x ☞Đặt t tan x ecot x ecot x I 4 dx dx . dx 2 ⑨ Cú t cot x 1 cos2x 2sin x sin2 x 6 ☞Đặt t cot x Ⓑ BÀI TẬP RẩN LUYỆN x2 4x 3 khi x 1 4 f tan x Cõu 1: Cho hàm số f x . Gớa trị của dx bằng x 1 2 0 cos x 3x 1 khi x 1 Ⓐ 1 Ⓑ 1 Ⓒ Ⓓ . 2. . ― 2. . 1. . ― 1. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 Cõu 2: Cho hàm số ( ) = | 2 ― 3 + 2|. Giỏ trị của f 2cos x 1 sin xdx bằng 0 Ⓐ 1 Ⓑ 1 Ⓒ Ⓓ . 2. . ― 2. . 1. . ― 1. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... .........................................................................................................................................................
- ......................................................................................................................................................... 푙푛 ℎ푖 ≥ 1 1 1 4 Cõu 3: Cho hàm số ( ) = 2 . Giỏ trị của ∫ ( 표푡 ) ― ∫ ( ) 3 ― 2 ― 1 ℎ푖 < 1 ― 2 ―1 4 푠푖푛 bằng Ⓐ. 3. Ⓑ. 2. Ⓒ. ― 2. Ⓓ. 0. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 3x2 2 khi x 1 Cõu 4: Cho hàm số f x 1 . Biết khi x 1 x 2 2 1 f sin x cos x dx 3 f 3 2x dx a bln 2, với , là cỏc số nguyờn. Giỏ trị của + 15 0 0 bằng Ⓐ. 18.Ⓑ. 10.Ⓒ. 48.Ⓓ. 6. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 4 푠푖푛 8 표푠 1 Cõu 5: Cho ( ) = | 2 ― 3 + 2|. Giỏ trị của ∫2 = + 푙푛 , với , là cỏc số 0 8 표푠 1 nguyờn. Giỏ trị của ― 2 bằng Ⓐ Ⓑ 4 Ⓒ 4 Ⓓ . 3. . ― 3. . 3. . ― 3. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ( + 2)푒 khi ≥ 0 1 4 Cõu 6: Cho hàm số ( ) = 2 . Giỏ trị của ∫ (푡 푛 ) bằng + 2khi < 0 ― 2 4 표푠 10 3 1 Ⓐ Ⓑ Ⓒ Ⓓ . 3 . . 7. . 7. . 7. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... .........................................................................................................................................................
- ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ( + 1)푙푛 ℎ푖 ≥ 1 2 2 Cõu 7: Cho hàm số ( ) = 4( ― 1) ℎ푖 < 1 . Giỏ trị của ∫ 표푡 (푠푖푛 ) . bằng 4 Ⓐ. 1 ― 2푙푛 2. Ⓑ. 1 ― 푙푛 2. Ⓒ. 푙푛 2. Ⓓ. 1. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 푒 khi ≥ 0 1 4 Cõu 8: Cho hàm số ( ) = 2 . Giỏ trị của ∫ (2 푡 푛 + 1) bằng 2 + khi < 0 ― 2 4 표푠 5 5 7 7 Ⓐ. 3 Ⓑ. 3 Ⓒ. 3 Ⓓ. 푒 + 12. 2푒 + 6. 4푒 ― 6. ― 6. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ( + 2)푒 khi ≥ 0 Cõu 9: Cho hàm số ( ) = 2 + khi < 0 cú đạo hàm trờn ℝ (với , là cỏc số thực). 1 4 Biết ∫ ( 표푡 ) = ―2푒 ― 5. Giỏ trị của 2 ― bằng ― 2 4 푠푖푛 Ⓐ. 10. Ⓑ. 18. Ⓒ. ―14. Ⓓ. ― 18. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ............................................................................................................................................. ......................................................................................................................................................... ......................................................................................................................................................... x 4 x 1 khi x 0 Cõu 10: Cho hàm số f x 1 . khi x 0 x 3 15 푙푛 50 44 3 3 표푡 2 표푠 Biết ∫ = , trong đú , , , là cỏc số nguyờn dương. Giỏ trị của 3 6 푠푖푛 45 + + + bằng Ⓐ. 9.Ⓑ. 10.Ⓒ. 11.Ⓓ. 12.
- Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 푒2 ℎ푖 ≥ 0 1 푒2 Cõu 11: Cho hàm số ( ) = . Biết tớch phõn ∫ ( ) = + ( là phõn số 2 + + 2 ℎ푖 < 0 ―1 tối giản). Giỏ trị + + bằng Ⓐ. 7.Ⓑ. 8.Ⓒ. 9.Ⓓ. 10. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 (1 + ) ℎ푖 ≥ 3 푒4 ( 푙푛 ) Cõu 12: Cho hàm số ( ) = 1 . Tớch phõn ∫ bằng: ℎ푖 < 3 푒2 4 40 95 189 189 Ⓐ. .Ⓑ. .Ⓒ. .Ⓓ. . 3 ― 푙푛 2 6 + 푙푛 2 4 + 푙푛 2 4 ― 푙푛 2 Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 1 ℎ푖 ≥ 1 1 Cõu 13: Cho hàm số ( ) = . Tớch phõn ∫ (3 1 ― ) = ( là phõn số tối giản), + 1 ℎ푖 < 1 ―2 푛 푛 khi đú ― 2푛 bằng: Ⓐ. 1.Ⓑ. 2.Ⓒ. 3.Ⓓ. 4. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... Cõu 14: Cho hàm số liờn tục trờn ℝ và 1 = 4, 3 = 6. Tớnh = 1 ( ) ∫0 ( ) ∫0 ( ) ∫―1 (|2 + 1|) Ⓐ. = 3.Ⓑ. = 5.Ⓒ. = 6.Ⓓ. = 4. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... .........................................................................................................................................................
- ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... Cõu 15: Cho 퐹( ) là một nguyờn hàm của hàm số ( ) = |1 + | ― |1 ― | trờn tập ℝ và thỏa món 퐹(1) = 3. Tớnh tổng 퐹(0) +퐹(2) +퐹( ―3). Ⓐ. 8.Ⓑ. 12.Ⓒ. 14.Ⓓ. 10. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 5 Cõu 16: Biết = ∫ 2| 2| 1 = 4 + 푙푛 2 + 푙푛 5 với , ∈ ℤ. Tớnh 푆 = + . 1 Ⓐ. 푆 = 9.Ⓑ. 푆 = 11.Ⓒ. 푆 = ―3.Ⓓ. 푆 = 5. Lờ Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... Cõu 17: Cho hàm số ( ) cú đạo hàm liờn tục trờn ℝ thỏa món ( 3 + 3 + 1) = 3 + 2, với mọi ∈ ℝ 5 .Tớch phõn ′ bằng ∫1 ( ) Ⓐ 31 Ⓑ 17 Ⓒ 33 Ⓓ 49 . ― 4 . . 4 . . 4 . . 4 . Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... Cõu 18: Cho hàm số = ( ) xỏc định và liờn tục trờn ℝ thoả ( 5 + 4 + 3) = 2 + 1,∀ ∈ ℝ. Tớch phõn 8 bằng ∫―2 ( ) Ⓐ Ⓑ Ⓒ 32 Ⓓ . 2. . 10. . 3 . . 72. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... .........................................................................................................................................................
- ......................................................................................................................................................... Cõu 19: Cho hàm số = ( ) xỏc định và liờn tục trờn ℝ thỏa món 2[ ( )]3 +3 ( ) + 5 = với ∀ ∈ ℝ. Tớnh = 10 . ∫5 ( ) Ⓐ. = 0.Ⓑ. = 3.Ⓒ. = 5.Ⓓ. = 6 Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 1 2 Cõu 20: Cho hàm số ( ) xỏc định ℝ\ , thỏa ′( ) = , (0) = 1 và (1) = 2. Giỏ trị của biểu thức 2 2 1 ( ―1) + (3) bằng Ⓐ. 풍풏 . Ⓑ. + 풍풏 . Ⓒ. + 풍풏 . Ⓓ. 4 + 푙푛 15. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 3 + 2 khi ≥ 0 2 Cõu 21: Cho hàm số ( ) = . Khi đú = ∫ 표푠 (푠푖푛 ) bằng 5 ― khi < 0 ― 2 15 17 Ⓐ. .Ⓑ. .Ⓒ. .Ⓓ. . 2 15 8 2 Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 ― 2 + 3 khi ≥ 2 1 Cõu 22: Cho hàm số ( ) = . Khi đú = ∫ (3 ― 2 ) bằng + 1 khi < 2 0 Ⓐ 41 Ⓑ Ⓒ 41 Ⓓ 41 . 2 . . 21. . 12. . 21. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 3 + 2 khi ≥ Cõu 23: Cho hàm số ( ) = 2. Khi đú = ∫2 푠푖푛 ( 표푠 + 1) bằng ― 2 khi < 3 0 2
- 35 19 10 Ⓐ. .Ⓑ. .Ⓒ. .Ⓓ. . 12 3 4 3 Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 ― khi ≥ 0 2 Cõu 24: Cho hàm số ( ) = . Khi đú = ∫ 표푠 (푠푖푛 ) bằng khi < 0 ― 2 Ⓐ 2 Ⓑ Ⓒ 1 Ⓓ 4 . ― 3. . ―1. . ― 3. . ― 3. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 + + 1 khi ≥ 3 2 Cõu 25: Cho hàm số ( ) = . Khi đú = ∫ ( 2 + 1) bằng 2 ― 1 khi < 3 0 Ⓐ Ⓑ 73 Ⓒ 74 Ⓓ . 24. . 3 . . 3 . . 25. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 1 3 + 3khi < Cõu 26: Cho hàm số ( ) = 2. Tớnh tớch phõn ∫2 (푠푖푛 ) 표푠 . + 4khi ≥ 1 0 2 Ⓐ Ⓑ 17 Ⓒ 13 Ⓓ 21 . 8. . 4 . . 2 . . 5 . Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... .........................................................................................................................................................
- ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 2 + 1khi ≥ 0 Cõu 27: Cho hàm số ( ) = . Tớnh tớch phõn ∫3 (3 표푠 ― 2)푠푖푛 . 2 2 ― + 1khi < 0 0 33 15 19 Ⓐ. .Ⓑ. .Ⓒ. 12.Ⓓ. . 2 23 24 Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 1 ― khi ≤ 1 4 Cõu 28: Cho hàm số ( ) = . Tớnh tớch phõn ∫ (5 푠푖푛 2 ― 1) 표푠 2 . 2 ― 2khi > 1 ― 2 Ⓐ 11 Ⓑ 43 Ⓒ 31 Ⓓ 31 . 10. . 31. . 30. . 10. Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 2 3 ― ― 5khi ≥ 2 푒 1 Cõu 29: Cho hàm số ( ) = . Tớnh tớch phõn ∫1 (2 + 푙푛 ) . 11 ― khi < 2 푒 69 25 Ⓐ. .Ⓑ. .Ⓒ. .Ⓓ. . 2 12 2 30 Lời giải ..................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... ......................................................................................................................................................... 1 ― 2khi ≤ 3 푙푛 2 Cõu 30: Cho hàm số ( ) = . Tớnh tớch phõn ∫ (3푒 ― 1)푒 . 7 ― 5 khi > 3 0 13 102 94 25 Ⓐ. .Ⓑ. .Ⓒ. .Ⓓ. . 15 ― 33 ― 9 9 Lời giải
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