Tài liệu ôn thi Đại học môn Toán chuẩn

Nguyen Van Linh-THPT GIAO THUY C 1 Chƣơng IV. Nguyên hàm, tích phân và ứng dụng §1. Nguyên hàm 1. Khái niệm a) ĐN. Cho hàm số 𝒇 𝒙 . Hàm số 𝑭(𝒙) gọi là một nguyên hàm của 𝒇(𝒙) nếu 𝑭′ 𝒙 = 𝒇 𝒙 và ta viết 𝒇(𝒙) 𝒅𝒙 = 𝑭 𝒙 + 𝑪, 𝑪 ∈ 𝑹. VD. i) 3𝑥2 𝑑𝑥 = 𝑥3 + 𝐶; 1 2 𝑥 𝑑𝑥 = 𝑥 + 𝐶. ii) cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶 ; 1 cos 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶. iii) 𝑒𝑥𝑑𝑥 = 𝑒𝑥 + 𝐶; 1 𝑥 𝑑𝑥 = ln 𝑥 + 𝐶. b) Bảng nguyên hàm 𝒅𝒙 = 𝒙 + 𝑪; 𝒙𝜶𝒅𝒙 = 𝒙𝜶+𝟏 𝜶+𝟏 + 𝑪 𝜶 ≠ −𝟏 ; 𝟏 𝒙 𝒅𝒙 = 𝒍𝒏 𝒙 + 𝑪 ; 𝟏 𝒙𝜶 𝒅𝒙 = 𝒙−𝜶𝒅𝒙 𝜶 ≠ 𝟏 . 𝒔𝒊𝒏 𝒙𝒅𝒙 = − 𝒄𝒐𝒔 𝒙 + 𝑪; 𝒄𝒐𝒔 𝒙 𝒅𝒙 = 𝒔𝒊𝒏 𝒙 + 𝑪. 𝟏 𝒄𝒐𝒔𝟐 𝒙 𝒅𝒙 = 𝒕𝒂𝒏 𝒙 + 𝑪; 𝟏 𝒔𝒊𝒏𝟐 𝒙 𝒅𝒙 = − 𝒄𝒐𝒕𝒙 + 𝑪. 𝒆𝒙𝒅𝒙 = 𝒆𝒙 + 𝑪; 𝒂𝒙𝒅𝒙 = 𝒂𝒙 𝒍𝒏𝒂 + 𝑪 𝟎 < 𝑎 ≠ 1 . c) Tính chất (𝒂𝒇 𝒙 + 𝒃𝒈 𝒙 ) 𝒅𝒙 = 𝒂 𝒇(𝒙) 𝒅𝒙 + 𝒃 𝒈(𝒙) 𝒅𝒙. * Đa thức, phân thức, căn thức 1. a) 3𝑥2 − 2 𝑥3 2 𝑑𝑥 = (9𝑥4 − 12 𝑥 + 4 𝑥6 ) 𝑑𝑥 = 9 5 𝑥5 − 12 𝑙𝑛 |𝑥| − 4 5𝑥5 + 𝐶. b) 𝑥 𝑥 + 1 (𝑥 + 2) 𝑑𝑥 = (𝑥3 + 3𝑥2 + 2𝑥) 𝑑𝑥 = 𝑥4 4 + 𝑥3 + 𝑥2 + 𝐶. c) ( 5 𝑥 − 1 𝑥3 4 )𝑑𝑥 = (5𝑥 − 1 2 − 𝑥− 3 4) 𝑑𝑥 = 10 𝑥 − 4 𝑥 4 + 𝐶. d) 𝑥 𝑥 𝑥 𝑑𝑥 = 𝑥 7 8 𝑑𝑥 = 8 15 𝑥15 8 + 𝐶. e) 𝑥 − 1 (𝑥 − 𝑥 + 1) 𝑑𝑥 = (𝑥 𝑥 + 1) 𝑑𝑥 = 2 5 𝑥5 + 𝑥 + 𝐶. f) (𝑥 − 1 𝑥 ) 𝑥4 𝑥 3 𝑑𝑥 = (𝑥 5 2 − 𝑥 1 2) 𝑑𝑥 = 2 7 𝑥7 − 2 3 𝑥3 + 𝐶. g) 𝑥−13𝑥2+𝑥2𝑒𝑥 𝑥2 𝑑𝑥 = (𝑥 − 3 2 − 13 + 𝑒𝑥 ) 𝑑𝑥 = − 2 𝑥 − 13𝑥 + 𝑒𝑥 + 𝐶. h) 𝑥4+𝑥−4+2 𝑥3 𝑑𝑥 = 𝑥4+1 𝑥5 𝑑𝑥 = 𝑙𝑛 𝑥 − 1 4𝑥4 + 𝐶. * Lƣợng giác 2. a) 𝑡𝑎𝑛2 𝑥 𝑑𝑥 = (𝑡𝑎𝑛2 𝑥 + 1 − 1) 𝑑𝑥 = 𝑡𝑎𝑛 𝑥 − 𝑥 + 𝐶. b) 𝑐𝑜𝑡2 𝑥 𝑑𝑥 = − 𝑐𝑜𝑡 𝑥 − 𝑥 + 𝐶. c) (4 𝑠𝑖𝑛 𝑥 2 𝑐𝑜𝑠 𝑥 2 + 𝑡𝑎𝑛2 𝑥) 𝑑𝑥 = 2 𝑠𝑖𝑛 𝑥 + 1 𝑐𝑜𝑠 2 𝑥 − 1 𝑑𝑥 = − 2 𝑐𝑜𝑠 𝑥 + 𝑡𝑎𝑛 𝑥 − 𝑥 + 𝐶. d) 𝑠𝑖𝑛 3 𝑥−2 3 𝑠𝑖𝑛 2 𝑥 𝑑𝑥 = ( 𝑠𝑖𝑛 𝑥 3 − 2 3 𝑠𝑖𝑛 2 𝑥 ) 𝑑𝑥 = − 1 3 𝑐𝑜𝑠 𝑥 + 2 3 𝑡𝑎𝑛 𝑥 + 𝐶. e) 1+𝑐𝑜𝑠 2 𝑥 1+𝑐𝑜𝑠 2𝑥 𝑑𝑥 = 1 2 ( 1 𝑐𝑜𝑠 2 𝑥 + 1) 𝑑𝑥 = 1 2 𝑡𝑎𝑛 𝑥 + 𝑥 2 + 𝐶. f) (𝑐𝑜𝑠2 𝑥 2 + 𝑥) 𝑑𝑥 = ( 1+𝑐𝑜𝑠 𝑥 2 + 𝑥) 𝑑𝑥 = 𝑥 2 + 𝑠𝑖𝑛 𝑥 2 + 𝑥2 2 + 𝐶. * Mũ, loga 3. a) 3𝑥𝑒𝑥 𝑑𝑥 = 3𝑒 𝑥 𝑑𝑥 = 3𝑒 𝑥 1+𝑙𝑛 3 + 𝐶. b) 𝑒𝑥 2𝑥 𝑑𝑥 = 𝑒 2 𝑥 𝑑𝑥 = 𝑒 2 𝑥 1+𝑙𝑛 2 + 𝐶. c) 3.2𝑥−2.3𝑥 2𝑥 𝑑𝑥 = (3 − 2 3 2 𝑥 ) 𝑑𝑥 = 3𝑥 − 2.3𝑥 𝑙𝑛 3 2 .2𝑥 + 𝐶. d) 2𝑥 + 3𝑥 2 𝑑𝑥 = (4𝑥 + 2.6𝑥 + 9𝑥 ) 𝑑𝑥 = 4𝑥 𝑙𝑛 4 + 2 6𝑥 𝑙𝑛 6 + 9𝑥 𝑙𝑛 9 + 𝐶. e) 2𝑥+1−6𝑥−1 10𝑥 𝑑𝑥 = (2 1 5 𝑥 − 1 6 3 5 𝑥 ) 𝑑𝑥 = − 2 5𝑥 𝑙𝑛 5 − 3𝑥 6.𝑙𝑛 3 5 .5𝑥 + 𝐶. f) (2𝑒𝑥 − 3 𝑥 + 𝑥) 𝑑𝑥 = 2𝑒𝑥 − 3 𝑙𝑛 |𝑥| + 𝑥2 2 + 𝐶. 2. Nguyên hàm của hàm hợp a) Vi phân 𝒅𝒇 𝒙 = 𝒇′ 𝒙 𝒅𝒙. VD. Tính i) 𝑑 𝑥2 − 2 tan 𝑥 + 3𝑒𝑥 = 2𝑥 − 2 cos 2 𝑥 + 3𝑒𝑥 . ii) 𝑑 𝑥2 − 𝑥3 − 2 = 2𝑥 − 3𝑥2 𝑑𝑥. iii) 𝑑 sin 𝑢 2 cos 𝑢 2 = 𝑑 sin 𝑢 2 = cos 𝑢 2 𝑑𝑢. iv) 𝑑 sin2 𝑡 = 2 sin 𝑡 cos 𝑡 𝑑𝑡 = sin 2𝑡 𝑑𝑡. b) ĐL: Nếu 𝒇 𝒙 𝒅𝒙 = 𝑭 𝒙 + 𝑪 và 𝒖 = 𝒖(𝒙) thì 𝒇 𝒖 𝒙 𝒅𝒖 = 𝑭 𝒖 𝒙 + 𝑪. c) Bảng nguyên hàm của hàm hợp 𝒖𝜶𝒅𝒖 = 𝒖𝜶+𝟏 𝜶+𝟏 + 𝑪 𝜶 ≠ −𝟏 ; 𝒅𝒖 𝒖 = 𝒍𝒏 𝒖 + 𝑪. 𝒔𝒊𝒏 𝒖 𝒅𝒖 = − 𝒄𝒐𝒔 𝒖 + 𝑪; 𝒄𝒐𝒔𝒖𝒅𝒖 = 𝒔𝒊𝒏 𝒖 + 𝑪. 𝒅𝒖 𝒄𝒐𝒔𝟐 𝒖 = 𝒕𝒂𝒏 𝒖 + 𝑪; 𝒅𝒖 𝒔𝒊𝒏𝟐 𝒖 = − 𝒄𝒐𝒕𝒖 + 𝑪. 𝒆𝒖𝒅𝒖 = 𝒆𝒖 + 𝑪; 𝒂𝒖𝒅𝒖 = 𝒂𝒖 𝒍𝒏 𝒂 + 𝑪 𝟎 < 𝑎 ≠ 1 . * Đa thức, phân thức, căn thức Nguyen Van Linh-THPT GIAO THUY C 2 4. a) 𝑥2 𝑥−3 𝑑𝑥 = (𝑥 + 3 + 9 𝑥−3 ) 𝑑𝑥 = 𝑥2 2 + 3𝑥 + 9 𝑙𝑛 |𝑥 − 3| + 𝐶. b) 𝑥 1 + 𝑥 𝑑𝑥 = (1 + 𝑥 − 1) 1 + 𝑥𝑑𝑥 = 2 5 1 + 𝑥 5 − 2 3 1 + 𝑥 3 + 𝐶. c) 1+𝑥− 1−𝑥 1−𝑥2 𝑑𝑥 = 1 1−𝑥 − 1 1+𝑥 𝑑𝑥 = −2 1 − 𝑥 − 2 1 + 𝑥 + 𝐶. d) 𝑑𝑥 𝑥+ 𝑥+1 = ( 𝑥 + 1 − 𝑥) 𝑑𝑥 = 2 3 𝑥 + 1 3 − 2 3 𝑥3 + 𝐶. e) 2𝑥𝑑𝑥 𝑥+ 𝑥2−1 = (2𝑥2 − 2𝑥 𝑥2 − 1)𝑑𝑥 = 2 3 𝑥3 − 2 3 𝑥2 − 1 3 + 𝐶. f) 𝑥𝑑𝑥 4−𝑥+ 4+𝑥 = 1 2 ( 4 − 𝑥 + 4 + 𝑥) 𝑑𝑥 = 1 3 4 + 𝑥 3 + 1 3 4 − 𝑥 3 + 𝐶. * Lƣợng giác 5. a) 4𝑑𝑥 𝑠𝑖𝑛2 𝑥 𝑐𝑜𝑠 2 𝑥 = 16𝑑𝑥 𝑠𝑖𝑛 2 2𝑥 = −8 𝑐𝑜𝑡 2𝑥 + 𝐶. b) 𝑐𝑜𝑠 2𝑥 𝑠𝑖𝑛 2 𝑥 𝑐𝑜𝑠 2 𝑥 𝑑𝑥 = 4𝑐𝑜𝑠 2𝑥 𝑠𝑖𝑛 2 2𝑥 𝑑𝑥 = − 2 𝑠𝑖𝑛 2𝑥 + 𝐶. c) 𝑡𝑎𝑛3 𝑥 𝑑𝑥 = 1 2 𝑡𝑎𝑛2 𝑥 + 𝑙𝑛 | 𝑐𝑜𝑠 𝑥 | + 𝐶. d) 𝑡𝑎𝑛4 𝑥 𝑑𝑥 = 1 3 𝑡𝑎𝑛3 𝑥 − 𝑡𝑎𝑛 𝑥 + 𝑥 + 𝐶. e) 𝑡𝑎𝑛5 𝑥 𝑑𝑥 = 1 4 𝑡𝑎𝑛4 𝑥 − 1 2 𝑡𝑎𝑛2 𝑥 − 𝑙𝑛 | 𝑐𝑜𝑠 𝑥 | + 𝐶. f) 𝑡𝑎𝑛6 𝑥 𝑑𝑥 = 1 5 𝑡𝑎𝑛5 𝑥 − 1 3 𝑡𝑎𝑛3 𝑥 + 𝑡𝑎𝑛 𝑥 − 𝑥 + 𝐶. g) 𝑠𝑖𝑛 2𝑥 𝑐𝑜𝑠 8𝑥 𝑑𝑥 = 1 12 𝑐𝑜𝑠 6𝑥 − 1 20 𝑐𝑜𝑠 10𝑥 + 𝐶. h) 𝑠𝑖𝑛 𝑥 𝑠𝑖𝑛 2𝑥 𝑠𝑖𝑛 3𝑥 𝑑𝑥 = − 1 16 𝑐𝑜𝑠 4𝑥 − 1 8 𝑐𝑜𝑠 2𝑥 + 1 24 𝑐𝑜𝑠 6𝑥 + 𝐶. i) 𝑐𝑜𝑠 𝑥+𝑐𝑜𝑠 2𝑥+𝑐𝑜𝑠 3𝑥 𝑠𝑖𝑛 𝑥+𝑠𝑖𝑛 2𝑥+𝑠𝑖𝑛 3𝑥 𝑑𝑥 = 1 2 𝑙𝑛 | 𝑠𝑖𝑛 2𝑥 | + 𝐶. j) 𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥 𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥 𝑑𝑥 = 𝑙𝑛 𝑠𝑖𝑛 𝑥 − 𝑐𝑜𝑠 𝑥 + 𝐶. * Mũ, loga 6. a) 𝑒𝑥𝑑𝑥 𝑒𝑥 +1 = 𝑑(𝑒𝑥 +1) 𝑒𝑥 +1 = 𝑙𝑛 𝑒𝑥 + 1 + 𝐶. b) 𝑒𝑥 −𝑒−𝑥 𝑒𝑥 +𝑒−𝑥 𝑑𝑥 = 𝑑(𝑒𝑥 +𝑒−𝑥 ) 𝑒𝑥 +𝑒−𝑥 = 𝑙𝑛(𝑒𝑥 + 𝑒−𝑥) + 𝐶. c) 𝑒𝑥𝑑𝑥 𝑒2𝑥 +4𝑒𝑥 +4 = 𝑒𝑥𝑑𝑥 𝑒𝑥 +2 = 𝑙𝑛 𝑒𝑥 + 2 + 𝐶. d) 𝑙𝑛 𝑥+1 2 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 + 1 2 𝑑 𝑙𝑛 𝑥 + 1 = 1 3 𝑙𝑛 𝑥 + 1 3 + 𝐶. e) 𝑠𝑖𝑛(𝑙𝑛 𝑥) 𝑥 𝑑𝑥 = 𝑠𝑖𝑛 𝑙𝑛 𝑥 𝑑 𝑙𝑛 𝑥 = − 𝑐𝑜𝑠(𝑙𝑛 𝑥) + 𝐶. f) 𝑙𝑛 𝑥+1 3 𝑥+1 𝑑𝑥 = 1 3 𝑙𝑛(𝑥 + 1) 𝑑 𝑙𝑛 𝑥 + 1 = 1 6 𝑙𝑛2(𝑥 + 1) + 𝐶. §2. Tích phân 1. Công thức Newton- Lepbnit 𝒇(𝒙) 𝒃 𝒂 𝒅𝒙 = 𝑭 𝒙 𝒂 𝒃 = 𝑭 𝒃 − 𝑭 𝒂 . 1. a)TN10. 𝐼 = 𝑥2 𝑥 − 1 2 1 0 𝑑𝑥 = (𝑥4 − 2𝑥3 + 1 0 𝑥2)𝑑𝑥 = 𝑥5 5 − 𝑥4 2 + 𝑥3 3 0 1 = 1 30 . b) TN08L2. 𝐼 = 6𝑥2 − 4𝑥 + 1 2 1 𝑑𝑥 = 2𝑥3 − 2𝑥2 + 𝑥 1 2 = 9. c) 𝐼 = ( 2. 𝑥 − 𝑥 3 + 1) 1 0 𝑑𝑥 = 2 2 3 + 1 4 . d) 𝐼 = 𝑥2+2 𝑥2 3 1 𝑑𝑥 = 10 3 . e) 𝐼 = 𝑠𝑖𝑛 𝑥 2 𝜋 2 0 𝑐𝑜𝑠 𝑥 2 𝑑𝑥 = 1 2 . f) 𝐼 = 𝑡𝑎𝑛2𝑥 𝜋 4 0 𝑑𝑥 = 1 − 𝜋 4 . g) h) 2. a) MTN09. 𝐼 = 𝑠𝑖𝑛 𝑥 2 + 𝑐𝑜𝑠 2𝑥 𝜋 2 0 𝑑𝑥 = 2 sin 𝑥 2 + cos 2𝑥 2 0 𝜋 2 = 2 − 2. b) TN08b. 𝐼 = 𝑥2 1 − 𝑥3 4 1 −1 𝑑𝑥 = − 1 3 1 − 1 −1 𝑥3 4𝑑(1 − 𝑥3) = − 1 15 1 − 𝑥3 5 −1 1 = 32 15 . c) TN07b. 𝐼 = 2𝑥𝑑𝑥 𝑥2+1 2 1 = 𝑑(𝑥2+1) 𝑥2+1 2 1 = 2 𝑥2 + 1 1 2 = 2 5 − 2 . d) D05. 𝐼 = 𝑒𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠 𝑥 𝑑𝑥 = 𝜋 2 0 𝑒𝑠𝑖𝑛 𝑥𝑑𝑥 + 𝜋 2 0 𝑐𝑜𝑠2𝑥 𝜋 2 0 𝑑𝑥 = 𝑒𝑠𝑖𝑛 𝑥 + 𝑥 2 + 𝑠𝑖𝑛 2𝑥 4 0 𝜋 2 = 𝑒 + 𝜋 4 − 1. e) DBB05. 𝐼 = 𝑡𝑎𝑛 𝑥 + 𝑒𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 𝑥 𝜋 4 0 𝑑𝑥 = 𝑡𝑎𝑛 𝑥 𝜋 4 0 𝑑𝑥 + 𝑒𝑠𝑖𝑛 𝑥𝑐𝑜𝑠 𝑥 𝜋 4 0 𝑑𝑥 = 𝑙𝑛 𝑐𝑜𝑠 𝑥 + 𝑒𝑠𝑖𝑛 𝑥 0 𝜋 4 = 1 2 ln 2 + 𝑒1/ 2 − 1. f) B03. 𝐼 = 1−2 𝑠𝑖𝑛 2 𝑥 1+𝑠𝑖𝑛 2𝑥 𝜋 4 0 𝑑𝑥 = cos 2𝑥 1+sin 2𝑥 𝜋 4 0 𝑑𝑥 = 1 2 𝑑(1+sin 2𝑥) 1+sin 2𝑥 𝜋 4 0 = 1 2 ln |1 + sin 2𝑥 | 0 𝜋 4 = 1 2 ln 2. g) h) * Tích phân hàm phân thức và phƣơng pháp đồng nhất hệ số 𝒅𝒙 𝒙+𝒂 = 𝒍𝒏 𝒙 + 𝒂 + 𝑪, 𝒙𝒅𝒙 𝒙𝟐+𝒂 = 𝟏 𝟐 𝒍𝒏 𝒙𝟐 + 𝒂 + 𝑪. 3. a) Tìm 𝐴 và 𝐵 → 5𝑥−3 𝑥2−3𝑥+2 = 𝐴 𝑥−1 + 𝐵 𝑥−2 . Tính 𝐼 = 5𝑥−3 𝑥2−3𝑥+2 4 3 𝑑𝑥. G: 𝐼 = (− 2 𝑥−1 + 7 𝑥−2 ) 4 3 𝑑𝑥 = −2 𝑙𝑛 𝑥 − 1 + 7 𝑙𝑛 𝑥 − 2 3 4 = 9 𝑙𝑛 2 − 2 𝑙𝑛 3. b) Tìm 𝐴, 𝐵 và 𝐶 → 2𝑥+3 𝑥3+𝑥2−2𝑥 = 𝐴 𝑥 + 𝐵 𝑥−1 + 𝐶 𝑥+2 . Tính 𝐼 = 2𝑥+3 𝑥3+𝑥2−2𝑥 𝑑𝑥. G: 𝐼 = (− 3 2𝑥 + 5 3 𝑥−1 − 1 6(𝑥+2) ) 𝑑𝑥 = − 3 2 𝑙𝑛 |𝑥| + 5 3 𝑙𝑛 |𝑥 − 1| − 1 6 𝑙𝑛 |𝑥 + 2| + 𝐶. c) Tìm 𝐴, 𝐵 và 𝐶 → 𝑥2+4𝑥+2 𝑥+1 3 = 𝐴 𝑥+1 + 𝐵 𝑥+1 2 + 𝐶 𝑥+1 3 . Tính 𝐼 = 𝑥2+4𝑥+2 𝑥+1 3 𝑑𝑥. G: 𝐼 = ( 1 𝑥+1 + 2 𝑥+1 2 − 1 𝑥+1 3 ) 𝑑𝑥 = 𝑙𝑛 𝑥 + 1 − 2 𝑥+1 + 1 2 𝑥+1 2 + 𝐶. d) Tìm 𝐴, 𝐵 và 𝐶 → 3𝑥2+3𝑥+3 𝑥3−3𝑥+2 = 𝐴 𝑥−1 2 + 𝐵 𝑥−1 + 𝐶 𝑥+2 . Tính 𝐼 = 3𝑥2+3𝑥+3 𝑥3−3𝑥+2 . Nguyen Van Linh-THPT GIAO THUY C 3 G: 𝐼 = ( 3 𝑥−1 2 + 2 𝑥−1 + 1 𝑥+2 ) 𝑑𝑥 = − 3 𝑥−1 + 2 𝑙𝑛 𝑥 − 1 + 𝑙𝑛 𝑥 + 2 + 𝐶. e) Tìm 𝐴, 𝐵 và 𝐶 → 𝑥2+2𝑥−2 𝑥3+1 = 𝐴 𝑥+1 + 𝐵𝑥+𝐶 𝑥2−𝑥+1 . Tính 𝐼 = 𝑥2+2𝑥−2 𝑥3+1 𝑑𝑥. G: 𝐼 = (− 1 𝑥+1 + 2𝑥−1 𝑥2−𝑥+1 ) 𝑑𝑥 = 𝑙𝑛 𝑥2 − 𝑥 + 1 − 𝑙𝑛 𝑥 + 1 + 𝐶. f) Tìm 𝐴, 𝐵 và 𝐶 → 𝑥2−𝑥−2 𝑥3−1 = 𝐴 𝑥−1 + 𝐵𝑥+𝐶 𝑥2+𝑥+1 . Tính 𝐼 = 𝑥2−𝑥−2 𝑥3−1 𝑑𝑥. G: 𝐼 = (− 1 𝑥−1 + 2𝑥+1 𝑥2+𝑥+1 ) 𝑑𝑥 = 𝑙𝑛 𝑥2 + 𝑥 + 1 − 𝑙𝑛 𝑥 − 1 + 𝐶. g) Tìm 𝐴 và 𝐵 → 𝑠𝑖𝑛 𝑥 𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥 = 𝐴 + 𝐵. 𝑐𝑜𝑠 𝑥−𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 𝑥+𝑠𝑖𝑛 𝑥 . Tính 𝐼 = 𝑠𝑖𝑛 𝑥𝑑𝑥 𝑐𝑜𝑠 𝑥+𝑠𝑖𝑛 𝑥 𝜋 2 0 . G: 𝐼 = ( 1 2 − 1 2 . 𝑐𝑜𝑠 𝑥−𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠 𝑥+𝑠𝑖𝑛 𝑥 ) 𝜋 2 0 𝑑𝑥 = 𝑥 2 − 1 2 𝑙𝑛 𝑐𝑜𝑠 𝑥 + 𝑠𝑖𝑛 𝑥 0 𝜋 2 = 𝜋 4 . 4. a) CD10. 𝐼 = 2𝑥−1 𝑥+1 1 0 𝑑𝑥. G: 𝐼 = (2 − 3 𝑥+1 ) 1 0 𝑑𝑥 = 2𝑥 − 3 𝑙𝑛 𝑥 + 1 0 1 = 2 − 3 𝑙𝑛 2. b) DBD02. 𝐼 = 𝑥3𝑑𝑥 𝑥2+1 1 0 . G: 𝐼 = (𝑥 − 𝑥 𝑥2+1 )𝑑𝑥 1 0 = 𝑥2 2 − 1 2 𝑙𝑛 𝑥2 + 1 0 1 = 1 2 + 1 2 𝑙𝑛 2. c) DBD07. 𝐼 = 𝑥 𝑥−1 𝑥2−4 1 0 𝑑𝑥. G: 𝐼 = (1 + 4−𝑥 𝑥2−4 ) 1 0 𝑑𝑥 = (1 + 𝑎 𝑥−2 + 𝑏 𝑥+2 ) 1 0 𝑑𝑥 = (1 + 1 2 𝑥−2 − 3 2(𝑥+2) ) 1 0 𝑑𝑥 = 𝑥 + 1 2 𝑙𝑛 𝑥 − 2 − 3 2 𝑙𝑛 𝑥 + 2 0 1 = 1 + 𝑙𝑛 2 − 3 2 𝑙𝑛 3. d) DBB04. 𝐼 = 𝑑𝑥 𝑥+𝑥3 3 1 . G: 𝐼 = ( 𝑎 𝑥 + 𝑏𝑥 +𝑐 1+𝑥2 ) 3 1 𝑑𝑥 = ( 1 𝑥 − 𝑥 1+𝑥2 ) 3 1 𝑑𝑥 = 𝑙𝑛 𝑥 − 1 2 𝑙𝑛 1 + 𝑥2 1 3 = 1 2 𝑙𝑛 3 2 . e) 𝐼 = 𝑥+5 𝑥3−4𝑥2+3𝑥 −1 −2 𝑑𝑥. G: 𝐼 = 5 3𝑥 − 3 𝑥−1 + 4 3(𝑥−3) −1 −2 𝑑𝑥 = 5 3 𝑙𝑛 𝑥 − 3𝑙𝑛 𝑥 − 1 + 4 3 𝑙𝑛 𝑥 − 3 −2 −1 = −2 𝑙𝑛 2 + 3 𝑙𝑛 3 − 4 3 𝑙𝑛 5. f) 𝐼 = 4𝑥−2 𝑥+2 𝑥2+1 1 0 𝑑𝑥. G: 𝐼 = (− 2 𝑥+2 + 2𝑥 𝑥2+1 ) 1 0 𝑑𝑥 = −2 𝑙𝑛 𝑥 + 2 + 𝑙𝑛 𝑥2 + 1 0 1 = 3 𝑙𝑛 2 − 2 𝑙𝑛 3. 2. Đổi biến số 𝒅𝒙 𝒙𝟐+𝒂𝟐 = 𝟏 𝒂 𝒂𝒓𝒄𝒕𝒂𝒏 𝒙 𝒂 + 𝑪, 𝒅𝒙 𝒂𝟐−𝒙𝟐 = 𝒂𝒓𝒄𝒔𝒊𝒏 𝒙 𝒂 + 𝑪. 5. a) 𝐼 = 𝑑𝑥 𝑥2+1 1 0 = 𝜋 4 , 𝐽 = 𝑑𝑥 𝑥2+3 1 0 = 𝜋 6 3 . b) DBA04. 𝐼 = 𝑥4−𝑥+1 𝑥2+4 2 0 𝑑𝑥 = 𝑥2 − 4 + 2 0 −𝑥+17 𝑥2+4 𝑑𝑥 = 𝑥2 − 4 − 𝑥 𝑥2+4 + 17 𝑥2+4 2 0 𝑑𝑥 = 𝑥3 3 − 4𝑥 − 1 2 ln 𝑥2 + 4 0 2 + 17 2 𝑡 0 𝜋 4 = 17 8 𝜋 − 16 3 − 1 2 𝑙𝑛 2. c) 𝐼 = 𝑥4+𝑥2−𝑥−3 𝑥3+𝑥 3 1 𝑑𝑥 = (𝑥 − 3 𝑥 + 3𝑥 𝑥2+1 − 3 1 1 𝑥2+1 ) 𝑑𝑥. d) 𝐼 = 𝑑𝑥 𝑥4+4𝑥2+3 1 0 = 𝜋 9−2 3 72 . Tích phân hàm căn thức Đặt 𝒕 = 𝝋 𝒙 𝒏ế𝒖 𝒇 = 𝒇 𝒙, 𝝋 𝒙 ; 6. a) DBB08. 𝐼 = 𝑥+1 4𝑥+1 2 0 𝑑𝑥 = 11 6 . b) DBA07. 𝐼 = 2𝑥+1 1+ 2𝑥+1 4 0 𝑑𝑥 = 2 + 𝑙𝑛 2. c) DBA06. 𝐼 = 𝑑𝑥 2𝑥+1+ 4𝑥+1 6 2 = 𝑙𝑛 3 2 − 1 12 . d) DBB06. 𝐼 = 𝑑𝑥 𝑥−2 𝑥−1 = 2 𝑙𝑛 2 + 1 10 5 . e) A04. 𝐼 = 𝑥 1+ 𝑥−1 𝑑𝑥 2 1 = 11 3 − 4 𝑙𝑛 2. 7. a) DBA08. 𝐼 = 𝑥𝑑𝑥 2𝑥+2 3 3 − 1 2 = 12 5 . b) DBA05. 𝐼 = 𝑥+2 𝑥+1 3 𝑑𝑥 = 231 10 . 7 0 c) DBB08. 𝐼 = 𝑥3𝑑𝑥 4−𝑥2 1 0 = 16 3 − 3 3. d) A03. 𝐼 = 𝑑𝑥 𝑥 𝑥2+4 2 3 5 = 1 4 𝑙𝑛 5 3 . e) DBA03. 𝐼 = 𝑥3 1 0 1 − 𝑥2𝑑𝑥 = 2 15 . f) 𝐼 = 𝑑𝑥 𝑥 𝑥2+1 8 3 = 1 2 ln 3 2 . g) 𝐼 = 𝑥3𝑑𝑥 1+𝑥2 3 7 0 = 141 10 . * Lƣợng giác 𝒕 = 𝒔𝒊𝒏 𝒙 𝒏ế𝒖 𝒇 = 𝒇 𝒔𝒊𝒏 𝒙 . 𝒄𝒐𝒔 𝒙; 𝒕 = 𝒄𝒐𝒔 𝒙 𝒏ế𝒖 𝒇 = 𝒇 𝒄𝒐𝒔 𝒙 . 𝒔𝒊𝒏 𝒙; 𝒕 = 𝒕𝒂𝒏 𝒙 𝒏ế𝒖 𝒇 = 𝒇 𝒕𝒂𝒏 𝒙 . 𝟏 𝒄𝒐𝒔𝟐𝒙 ; 𝒖 = 𝒔𝒊𝒏𝒙 ± 𝒄𝒐𝒔 𝒙 𝒏ế𝒖 𝒇 = 𝒇 𝒔𝒊𝒏 𝒙 ± 𝒄𝒐𝒔 𝒙, 𝒔𝒊𝒏 𝟐𝒙 ; 8. a) A09. 𝐼 = 𝑐𝑜𝑠3 𝑥 − 1 𝜋 2 0 𝑐𝑜𝑠2 𝑥 𝑑𝑥 HD: 𝐼 = 𝑐𝑜𝑠5 𝑥 𝑑𝑥 𝜋 2 0 − 𝑐𝑜𝑠2 𝑥 𝑑𝑥. 𝜋 2 0 Đặt 𝑡 = 𝑠𝑖𝑛 𝑥 → 𝐼 = 8 15 − 𝜋 4 . b) A08. 𝐼 = 𝑡𝑎𝑛 4 𝑥 𝑐𝑜𝑠 2𝑥 𝑑𝑥 𝜋 6 0 . HD: 𝐼 = 𝑡𝑎𝑛 4 𝑥 1−𝑡𝑎𝑛 2 𝑥 𝑐𝑜𝑠 2 𝑥 𝑑𝑥. Đặt 𝑡 = 𝑡𝑎𝑛 𝑥 → 𝐼 = 1 2 𝑙𝑛 2 + 3 − 10 9 3 . c) B08. 𝐼 = 𝑠𝑖𝑛 ⁡(𝑥− 𝜋 4 )𝑑𝑥 𝑠𝑖𝑛 2𝑥+2(1+𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥) 𝜋 4 0 . HD: 𝐼 = 1 2 (𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 𝑥)𝑑𝑥 𝑠𝑖𝑛 2𝑥+2(1+𝑠𝑖𝑛 𝑥+𝑐𝑜𝑠 𝑥) 𝜋 4 0 . Đặt 𝑡 = sin 𝑥 + cos 𝑥 → 𝐼 = 4−3 2 4 . d) DBA08. 𝐼 = 𝑠𝑖𝑛 2𝑥 𝑑𝑥 3+4 𝑠𝑖𝑛 𝑥−𝑐𝑜𝑠 2𝑥 𝜋 2 0 . HD: Đặt 𝑡 = 𝑠𝑖𝑛 𝑥 → 𝐼 = − 1 2 + 𝑙𝑛 2. e) B05. 𝐼 = 𝑠𝑖𝑛 2𝑥 𝑐𝑜𝑠 𝑥 1+𝑐𝑜𝑠 𝑥 𝜋 2 0 𝑑𝑥. HD: Đặt 𝑡 = 𝑐𝑜𝑠 𝑥 → 𝐼 = 2 𝑙𝑛 2 − 1. f) DBA05. 𝐼 = 𝑠𝑖𝑛2 𝑥 𝑡𝑎𝑛 𝑥 𝑑𝑥 𝜋 3 0 . HD: Đặt 𝑡 = 𝑐𝑜𝑠 𝑥 → 𝐼 = 𝑙𝑛 2 − 3 8 . g) 𝐼 = 𝑐𝑜𝑠 𝑥 𝑑𝑥 𝑠𝑖𝑛 2 𝑥−5 𝑠𝑖𝑛 𝑥+6 𝜋 3 𝜋 6 . Đặt 𝑡 = 𝑠𝑖𝑛 𝑥 → 𝐼 = 𝑙𝑛 3 6− 3 5 4− 3 . h) 𝐼 = 𝑠𝑖𝑛 3 𝑥𝑑𝑥 𝑐𝑜𝑠 𝑥+2 . 𝜋 3 0 Đặt 𝑡 = 𝑐𝑜𝑠 𝑥 → 𝐼 = 5 2 + 3 𝑙𝑛 5 6 . Nguyen Van Linh-THPT GIAO THUY C 4 i) 𝐼 = 𝑠𝑖𝑛 2𝑥𝑑𝑥 2 𝑠𝑖𝑛 2 𝑥+𝑐𝑜𝑠 2 𝑥 𝜋 6 0 . Đặt 𝑡 = 2 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠2 𝑥 → 𝐼 = 𝑙𝑛 5 4 . 9. a) A06. 𝐼 = 𝑠𝑖𝑛 2𝑥 𝑐𝑜𝑠 2 𝑥+4 𝑠𝑖𝑛 2 𝑥 𝜋 2 0 𝑑𝑥. HD: Đặt 𝑡 = 𝑐𝑜𝑠2 𝑥 + 4 𝑠𝑖𝑛2 𝑥 → 𝐼 = 2 3 . b) A05. 𝐼 = 𝑠𝑖𝑛 2𝑥+𝑠𝑖𝑛 𝑥 1+3 𝑐𝑜𝑠 𝑥 𝑑𝑥 𝜋 2 0 . HD: Đặt 𝑡 = 1 + 3 𝑐𝑜𝑠 𝑥 → 𝐼 = 34 27 . c) DBA02. 𝐼 = 1 − 𝑐𝑜𝑠3 𝑥 6 𝜋 2 0 . 𝑠𝑖𝑛 𝑥 𝑐𝑜𝑠5 𝑥 𝑑𝑥. HD: Đặt 𝑡 = 1 − 𝑐𝑜𝑠3 𝑥 6 → 𝐼 = 12 91 . 10. a) 𝐼 = 1 − 𝑥2 1 0 𝑑𝑥 = 𝜋 4 . b) 𝐼 = 𝑥2 1−𝑥2 2 2 0 𝑑𝑥 = 𝜋 8 − 1 4 . c) 𝐼 = 𝑥2 4 − 𝑥2 2 1 𝑑𝑥 = 1 2 ( 𝜋 3 + 3 8 ). d) 𝐼 = 3 − 𝑥2 3 3 2 0 𝑑𝑥 = 9 16 𝜋2 + 81 64 𝜋. * Mũ 𝒖 = 𝒆𝒙 𝒏ế𝒖 𝒇 = 𝒇 𝒆𝒙 𝒐𝒓 𝒖 = 𝒍𝒏 𝒙 𝒏ế𝒖 𝒇 = 𝒇 𝒍𝒏 𝒙 𝒙 . 11. a) A10. 𝐼 = 𝑥2+𝑒𝑥 +2𝑥2𝑒𝑥 1+2𝑒𝑥 1 0 𝑑𝑥. HD: 𝐼 = 𝑥2 + 𝑒𝑥 1+2𝑒𝑥 1 0 𝑑𝑥. Đặt 𝑡 = 𝑒𝑥 → 𝐼 = 1 3 + 1 2 𝑙𝑛 1+2𝑒 3 . b) D09. 𝐼 = 𝑑𝑥 𝑒𝑥 −1 3 1 . HD: Đặt 𝑡 = 𝑒𝑥 → 𝐼 = 𝑙𝑛 𝑒2 + 𝑒 + 1 − 2. c) B06. 𝐼 = 𝑑𝑥 𝑒𝑥 +2𝑒−𝑥−3 𝑙𝑛 5 𝑙𝑛 3 . HD: Đặt 𝑡 = 𝑒𝑥 → 𝐼 = 𝑙𝑛 3 2 . d) DBB02. 𝐼 = 𝑒𝑥 𝑑𝑥 𝑒𝑥 +1 3 𝑙𝑛 3 0 . HD: Đặt 𝑡 = 𝑒𝑥 + 1 → 𝐼 = 2 − 1. e) 𝐼 = 𝑑𝑥 𝑒𝑥 +2 𝑙𝑛 2 0 . Đặt 𝑡 = 𝑒𝑥 + 2 → 𝐼 = 1 2 2 𝑙𝑛 | 2− 2 ( 3+ 2) 2+ 2 ( 3− 2) |. 12. a) TN06b. 𝐼 = 𝑒𝑥 +1 𝑒𝑥 𝑒𝑥 −1 𝑙𝑛 5 𝑙𝑛 2 𝑑𝑥. HD: Đặt 𝑡 = 𝑒𝑥 − 1 → 𝐼 = 26 3 . b) DBB03. 𝐼 = 𝑒2𝑥𝑑𝑥 𝑒𝑥 −1 𝑙𝑛 5 𝑙𝑛 2 . HD: Đặt 𝑡 = 𝑒𝑥 − 1 → 𝐼 = 20 3 . c) DBD04. 𝐼 = 𝑒2𝑥 𝑒𝑥 + 1 𝑙𝑛 8 𝑙𝑛 3 𝑑𝑥. G: Đặt 𝑡 = 𝑒𝑥 + 1 → 𝐼 = 1076 15 . * Loga 13. a) B10. 𝐼 = 𝑙𝑛 𝑥 𝑥 2+𝑙𝑛 𝑥 2 𝑒 1 𝑑𝑥. HD: Đặt 𝑡 = 2 + 𝑙𝑛 𝑥 → 𝐼 = − 1 3 + 𝑙𝑛 3 2 . b) DBB06. 𝐼 = 3−2 𝑙𝑛 𝑥 𝑥 1+2 𝑙𝑛 𝑥 𝑒 1 𝑑𝑥. HD: Đặt 𝑡 = 1 + 2 𝑙𝑛 𝑥 → 𝐼 = 10 2−11 3 . c) DBD05. 𝐼 = 𝑙𝑛 2 𝑥 𝑥 𝑙𝑛 𝑥+1 𝑒3 1 𝑑𝑥. HD: Đặt 𝑡 = 𝑙𝑛 𝑥 + 1 → 𝐼 = 76 15 . d) B04. 𝐼 = 1+3 𝑙𝑛 𝑥 𝑙𝑛 𝑥 𝑥 𝑑𝑥 𝑒 1 . HD: Đặt 𝑡 = 1 + 3 𝑙𝑛 𝑥 → 𝐼 = 116 135 . e) 𝐼 = 1+𝑙𝑛 2 𝑥 𝑥 𝑒 1 𝑑𝑥. Đặt 𝑡 = 𝑙𝑛 𝑥 → 𝐼 = 4 3 . 14. i) 𝐼 = 𝑥2004 sin 𝑥 𝑑𝑥. 1 −1 Đặt 𝑡 = −𝑥 → 𝐼 = −𝐼 → 𝐼 = 0. ii) 𝐼 = cos 4 𝑥 cos 4 𝑥+sin 4 𝑥 𝜋 2 0 𝑑𝑥. Đặt 𝑡 = 𝜋 2 − 𝑥 → 𝐼 = sin 4 𝑡𝑑𝑡 sin 4 𝑡+cos 4 𝑡 𝜋 2 0 → 2𝐼 = 𝜋 2 → 𝐼 = 𝜋 4 . iii) 𝐼 = 𝑑𝑥 1+ 1+𝑥 0 −1 ; Đặt 𝑡 = 1 + 1 + 𝑥 → 𝐼 = 4𝑡2 − 4 2 1 𝑑𝑡 = 4 3 2 − 2 . iv) 𝐼 = 𝑑𝑥 cos 𝑥 sin 𝑥−cos 𝑥 𝜋 6 0 ; 𝐼 = 𝑑 tan 𝑥 tan 𝑥−1 𝜋 6 0 = ln tan 𝑥 − 1 0 𝜋 6 = ln 3− 3 3 . v) 𝐼 = 𝑑𝑥 1+𝑥+ 1+𝑥2 1 −1 = 1. vi) 𝐼 = 𝑒2𝑥𝑑𝑥 𝑒𝑥−1+ 𝑒𝑥 −2 ln 3 ln 2 = 2 ln 3 − 1. vii) 𝐼 = 𝑑𝑥 𝑥 cos 2(ln 𝑥+1) 𝑒4 𝑒−1 = tan 5. II. Dạng 3: Tích phân từng phần * sin, cos, tan 𝑷(𝒙) 𝒄𝒐𝒔 𝒙𝒅𝒙 = 𝑷 𝒙 𝒅 𝒔𝒊𝒏 𝒙 , 𝑷(𝒙) 𝒄𝒐𝒔𝟐𝒙 𝒅𝒙 = 𝑷(𝒙) 𝒅𝒕𝒂𝒏 𝒙. 15. a) TN08a. 𝐼 = (2𝑥 − 1) 𝜋 2 0 𝑐𝑜𝑠 𝑥 𝑑𝑥 = 2𝑥 − 𝜋 2 0 1 𝑑(sin 𝑥) = 𝜋 − 3. b) DBD07. 𝐼 = 𝑥2 𝑐𝑜𝑠 𝑥 𝑑𝑥 𝜋 2 0 = 𝑥2 𝜋 2 0 𝑑(𝑠𝑖𝑛 𝑥) = 𝜋2 4 − 2. 16. a) TN09. 𝐼 = 𝑥(1 + 𝑐𝑜𝑠 𝑥) 𝜋 0 𝑑𝑥 = 𝑥 𝑑(𝑥 + 𝜋 0 sin 𝑥) = 𝜋2 2 − 2. b) DBD06. 𝐼 = 𝑥 + 1 𝑠𝑖𝑛 2𝑥 𝑑𝑥 𝜋 2 0 = − 1 2 𝑥 + 𝜋 2 0 1 𝑑(𝑐𝑜𝑠 2𝑥) = 𝜋 4 + 1. c) DBD05. 𝐼 = 2𝑥 − 1 𝑐𝑜𝑠2 𝑥 𝑑𝑥 𝜋 2 0 = 2𝑥 − 𝜋 2 0 1 1+𝑐𝑜𝑠 2𝑥 2 𝑑𝑥 = 1 2 2𝑥 − 1 𝑑 𝜋 2 0 (𝑥 + 𝑠𝑖𝑛 2𝑥 2 ) = 𝜋2 8 − 𝜋 4 − 1 2 . d) D1D04. 𝐼 = 𝑥 𝜋2 0 𝑠𝑖𝑛 𝑥 𝑑𝑥 = −2 𝑥 𝜋2 0 𝑑(𝑐𝑜𝑠 𝑥) = 2𝜋2 − 8. e) DBA03. 𝐼 = 𝑥 1+𝑐𝑜𝑠 2𝑥 𝜋 4 0 𝑑𝑥 = 1 2 𝑥 𝑐𝑜𝑠 2 𝑥 𝜋 4 0 𝑑𝑥 = 1 2 𝑥 𝑑(𝑡𝑎𝑛 𝑥) 𝜋 4 0 = 1 2 𝑥 𝑡𝑎𝑛 𝑥 0 𝜋 4 − 𝑡𝑎𝑛 𝑥 𝑑𝑥 𝜋 4 0 = 1 2 𝜋 4 + ln cos 𝑥 0 𝜋 4 = 𝜋 8 + 1 2 ln 1 2 = 𝜋 8 − 1 4 ln 2. 17. i) 𝐼 = 𝑥2 + 1 sin 𝑥 𝜋 2 0 𝑑𝑥 = 𝜋 − 1. ii) 𝐼 = 𝑒2𝑥 𝜋 0 sin2 𝑥 𝑑𝑥 = 𝑒2𝜋−1 8 . iii) 𝐼 = 𝑥2 + 1 𝜋 2 0 sin 𝑥 𝑑𝑥 = 𝜋 − 1. iv) 𝐼 = 𝑥 sin 𝑥 cos 2 𝑥 𝜋 3 − 𝜋 3 𝑑𝑥 = 4𝜋 3 − 2 ln(2 + 3) * mũ 𝑷(𝒙) 𝒆𝒙𝒅𝒙 = 𝑷 𝒙 𝒅 𝒆𝒙 . 18. a) TN08L2b. 𝐼 = (4𝑥 + 1) 1 0 𝑒𝑥𝑑𝑥 = 4𝑥 + 1 0 1 𝑑(𝑒𝑥 ) = 𝑒 + 3. Nguyen Van Linh-THPT GIAO THUY C 5 b) TN06a. 𝐼 = 2𝑥 + 1 𝑒𝑥𝑑𝑥 1 0 = 2𝑥 + 1 𝑑(𝑒𝑥 ) 1 0 = 𝑒 + 1. c) DBD08. 𝐼 = 𝑥2 + 𝑥 + 1 1 0 𝑒𝑥𝑑𝑥 = 𝑥2 + 𝑥 + 1 0 1 𝑑(𝑒𝑥 ) = 2𝑒 − 2. 19. a) CD09. 𝐼 = 𝑒−2𝑥 + 𝑥 1 0 𝑒𝑥𝑑𝑥 = 𝑒−𝑥 1 0 𝑑𝑥 + 𝑥𝑒𝑥 1 0 𝑑𝑥 = 2 − 𝑒−1 . b) DBD08. 𝐼 = 𝑥𝑒2𝑥 − 𝑥 4−𝑥2 1 0 𝑑𝑥 = 𝑥𝑒2𝑥𝑑𝑥 1 0 − 𝑥 4−𝑥2 1 0 𝑑𝑥 = 3 + 𝑒2−7 4 . c) D06. 𝐼 = 𝑥 − 2 𝑒2𝑥𝑑𝑥 1 0 = 1 2 𝑥 − 2 1 0 𝑑(𝑒2𝑥) = 5−3𝑒2 4 . d) DBB04. 𝐼 = 𝑒𝑐𝑜𝑠 𝑥 𝜋 2 0 𝑠𝑖𝑛 2𝑥 𝑑𝑥 = −2 𝑐𝑜𝑠 𝑥 𝜋 2 0 𝑑 𝑒𝑐𝑜𝑠 𝑥 =. e) DBB03. Cho 𝑓 𝑥 = 𝑎 𝑥+1 3 + 𝑏𝑥𝑒𝑥 . 𝑎, 𝑏? → 𝑓 ′ 0 = −22 và 𝑓(𝑥) 1 0 𝑑𝑥 = 5. HD: 8,2 . f) DBD03. 𝐼 = 𝑥3𝑒𝑥 21 0 𝑑𝑥 = 1 2 𝑥2 1 0 𝑑(𝑒𝑥 2 ) = 1 2 . g) DBA02. 𝐼 = 𝑥 𝑒𝑥 + 𝑥 + 1 3 0 −1 𝑑𝑥 = 𝑥𝑒𝑥𝑑𝑥 0 −1 + 𝑥. 𝑥 + 1 3 𝑑𝑥 0 −1 = 3 4𝑒2 − 4 7 . * loga 𝒙𝜶𝒍𝒏 𝒙𝒅𝒙 = 𝒍𝒏 𝒙𝒅 𝒙𝜶+𝟏 𝜶+𝟏 . 20. a) TN07a. 𝐼 = 2𝑥 3 1 𝑙𝑛 𝑥𝑑𝑥 = ln 𝑥 𝑑(𝑥2) 3 1 = 9 ln 3 − 4. b) D08. 𝐼 = 𝑙𝑛 𝑥 𝑥3 𝑑𝑥 2 1 = ln 𝑥 2 1 𝑑( 𝑥−2 −2 ) = 3−2 ln 2 16 . c) DBD06. 𝐼 = 𝑥 − 2 𝑙𝑛 𝑥𝑑𝑥 2 1 = 𝑙𝑛 𝑥 2 1 𝑑( 𝑥2 2 − 2𝑥) = − ln 4 + 5 4 . d) DBB05. 𝐼 = 𝑥2 𝑒 1 𝑙𝑛 𝑥 𝑑𝑥 = 𝑙𝑛 𝑥 𝑒 1 𝑑( 𝑥3 3 ) = 2 9 𝑒3 + 1 9 . 21. a) D10. 𝐼 = 2𝑥 − 3 𝑥 𝑒 1 𝑙𝑛 𝑥 𝑑𝑥 = 2𝑥𝑙𝑛 𝑥 𝑒 1 𝑑𝑥 − 3 ln 𝑥 𝑥 𝑒 1 𝑑𝑥 = 𝑒2 2 − 1. b) B09. 𝐼 = 3+𝑙𝑛 𝑥 𝑥+1 2 3 1 𝑑𝑥 = 3 + 𝑙𝑛 𝑥 3 1 𝑑( −1 𝑥+1 ) = 3 4 + 1 4 ln 27 16 . c) D07. 𝐼 = 𝑥3 𝑙𝑛2 𝑥 𝑑𝑥 𝑒 1 = ln2 𝑥 𝑒 1 𝑑( 𝑥4 4 ) = 5𝑒4−1 32 . d) D04. 𝐼 = 𝑙𝑛 𝑥2 − 𝑥 𝑑𝑥 3 2 = 3 ln 3 − 2. e) DBD03. 𝐼 = 𝑥2+1 𝑥 𝑙𝑛 𝑥 𝑒 1 𝑑𝑥 = 𝑥 𝑒 1 ln 𝑥𝑑𝑥 + 𝑙𝑛 𝑥 𝑥 𝑒 1 𝑑𝑥 = 5+𝑒2 4 . f) 𝐼 = ln 1+𝑥 𝑥2 2 1 𝑑𝑥 = − 3 ln 3 2 + 3 ln 2. g) 𝐼 = (1 − 𝑥2) 𝑒 1 ln 𝑥 𝑑𝑥 = 8−2𝑒3 9 . §3. Ứng dụng của tích phân 1. Tính diện tích Dạng 1a. 𝑺 𝒚 = 𝒇 𝒙 , 𝑶𝒙, 𝒙 = 𝒂, 𝒙 = 𝒃 = 𝒇(𝒙) 𝒃 𝒂 𝒅𝒙 ∶ 𝒇 𝒙 ≥ 𝟎 1. 𝑆 𝑦 = 𝑥3, 𝑥 = 1, 𝑥 = 3, 𝑦 = 0 = 𝑥3 3 1 𝑑𝑥 = 20. 2. 𝑆 𝑦 = 𝑒𝑥 + 1, 𝑂𝑥, 𝑥 = 𝑙𝑛 3 , 𝑥 = 𝑙𝑛 8 = 𝑒𝑥 + 1 𝑙𝑛 8 𝑙𝑛 3 𝑑𝑥 = 2 + 𝑙𝑛 3 2 . 3. 𝑆 𝑦 = 𝑥 𝑙𝑛 2 𝑥2+1 𝑥2+1 , 𝑂𝑦, 𝑂𝑥, 𝑥 = 𝑒 − 1 = 𝑥 𝑙𝑛2(𝑥2+1) 𝑥2+1 𝑒−1 0 𝑑𝑥 = 1 6 . 4. 𝑆 𝑦 = 𝑙𝑛 𝑥 2 𝑥 , 𝑥 = 1, 𝑥 = 𝑒, 𝑦 = 0 = 𝑙𝑛 𝑥 2 𝑥 𝑒 1 𝑑𝑥 = 2 − 𝑒. Dạng 1b. 𝑺 𝒚 = 𝒇 𝒙 , 𝑶𝒙, 𝒙 = 𝒂, 𝒙 = 𝒃 = (−𝒇(𝒙)) 𝒃 𝒂 𝒅𝒙 ∶ 𝒇 𝒙 ≤ 𝟎 5. DBB02. 𝑚? ∈ 0, 5 6 → 𝑆 𝑦 = 1 3 𝑥3 + 𝑚𝑥2 − 2𝑥 − 2𝑚 − 1 3 , 𝑥 = 0, 𝑥 = 2, 𝑦 = 0 = 4. HD: 𝑆 = (− 1 3 𝑥3 − 𝑚𝑥2 + 2𝑥 + 2𝑚 + 1 3 ) 2 0 𝑑𝑥 = 4 → 𝑚 = 1 2 . Dạng 1c. 𝑺 𝒚 = 𝒇 𝒙 , 𝑶𝒙, 𝒙 = 𝒂, 𝒙 = 𝒃 = 𝒇 𝒙 𝒅𝒙 𝒃 𝒂 6. D03. 𝐼 = |𝑥2 − 𝑥|𝑑𝑥 2 0 = 1. 7. 𝑆 𝑦 = 𝑥2 − 2𝑥, 𝑥 = −1, 𝑥 = 2, 𝑦 = 0 = |𝑥2 − 2 −1 2𝑥|𝑑𝑥 = 8 3 . Dạng 1d. 𝑺 𝒙 = 𝒇 𝒚 , 𝑶𝒚, 𝒚 = 𝒂, 𝒚 = 𝒃 = |𝒇(𝒚)| 𝒃 𝒂 𝒅𝒚 8. 𝑆 𝑥 = 𝑠𝑖𝑛 𝑦, 𝑂𝑦, 𝑥 = 𝜋 3 , 𝑥 = 𝜋 4 = 𝑠𝑖𝑛 𝑦 𝜋 3 𝜋 4 𝑑𝑦 = 2−1 2 . Dạng 2a. 𝑺 𝒚 = 𝒇 𝒙 , 𝒚 = 𝒈 𝒙 = |𝒇 𝒙 − 𝒈(𝒙)| 𝒃 𝒂 𝒅𝒙 9. TN07L2a. 𝑆 𝑦 = −𝑥2 + 6𝑥, 𝑦 = 0 = | − 𝑥2 + 6 0 6𝑥|𝑑𝑥 = 36. 10. TN06. 𝑆 𝑦 = −𝑥3 + 3𝑥2, 𝑂𝑥 = | − 𝑥3 + 3𝑥2|𝑑𝑥 3 0 = 27 4 . 11. CD08. S 𝑦 = −𝑥2 + 4𝑥, 𝑦 = 𝑥 = | − 𝑥2 + 3 0 3𝑥| 𝑑𝑥 = 9 2 . 12. A07. S 𝑦 = 𝑒 + 1 𝑥, 𝑦 = 1 + 𝑒𝑥 𝑥 = |(𝑒𝑥 − 1 0 𝑒)𝑥| 𝑑𝑥 = 𝑒 2 − 1. 13. DBB07. S 𝑦 = 0, 𝑦 = 𝑥 1−𝑥 𝑥2+1 = | 𝑥(1−𝑥) 𝑥2+1 | 1 0 𝑑𝑥 = −1 + 𝜋 4 + 1 2 𝑙𝑛 2. 14. DBB07. S 𝑦 = 𝑥2, 𝑦 = 2 − 𝑥2 = |𝑥2 − 1 −1 2 − 𝑥2| 𝑑𝑥 = 𝜋 2 + 1 3 . 15. DBA06. S 𝑦 = 𝑥2 − 𝑥 + 3, 𝑦 = 2𝑥 + 1 = |𝑥2 − 3𝑥 + 2| 2 1 𝑑𝑥 = 1 6 . 16. B02. S 𝑦 = 4 − 𝑥2 4 , 𝑦 = 𝑥2 4 2 = | 4 − 𝑥2 4 − 2 2 −2 2 𝑥2 4 2 | 𝑑𝑥 = 2𝜋 + 4 3 . 17. DBD02. 𝑆 𝑦 = 1 3 𝑥3 − 2𝑥2 + 3𝑥, 𝑂𝑥 = | 1 3 𝑥3 − 3 0 2𝑥2 + 3𝑥| 𝑑𝑥 = 9 4 . Nguyen Van Linh-THPT GIAO THUY C 6 18. 𝑆 𝑦 = 𝑥 + 𝑠𝑖𝑛 𝑥 , 𝑦 = 𝑥, 𝑥 = 1, 𝑥 = 2𝜋 = 𝑠𝑖𝑛 𝑥 𝑑𝑥 2𝜋 1 = 1 + 𝑐𝑜𝑠 1. 19. 𝑆 𝑦 = 2𝑥 − 𝑥2, 𝑂𝑥 . 20. 𝑆 𝑦 = 𝑥2, 𝑦 = 4𝑥 − 3𝑥2 . 21. 𝑆 𝑦 = 𝑥, 𝑦 = 1 2 𝑥 . Dạng 2b. 𝑺 𝒚 = 𝒇 𝒙 , 𝒚 = 𝒈 𝒙 , 𝒙 = 𝒂, 𝒙 = 𝒃 = |𝒇 𝒙 − 𝒃 𝒂 𝒈(𝒙)| 𝒅𝒙 22. 𝑆 𝑦 = 1 𝑠𝑖𝑛 2 𝑥 , 𝑦 = 1 𝑐𝑜𝑠 2 𝑥 , 𝑥 = 𝜋 6 , 𝑥 = 𝜋 3 = | 1 𝑠𝑖𝑛 2 𝑥 − 𝜋 3 𝜋 6 1 𝑐𝑜𝑠 2 𝑥 | 𝑑𝑥 = 8 3 − 4. Dạng 2c. 𝑺 𝒚 = 𝒇 𝒙 , 𝒚 = 𝒈 𝒙 , 𝒙 = 𝒂 = |𝒇 𝒙 − 𝒈(𝒙)| 𝒃 𝒂 𝒅𝒙 23. D02. S 𝑦 = −3𝑥−1 𝑥−1 , 𝑂𝑥, 𝑂𝑦 = | −3𝑥−1 𝑥−1 | 0 − 1 3 𝑑𝑥 = −1 + 4 𝑙𝑛 4 3 . 24. 𝑆 𝑦 = 2𝑥 , 𝑦 = 3 − 𝑥, 𝑥 = 0 = 2𝑥 − 3 + 1 0 𝑥 𝑑𝑥 = 5 2 − 1 𝑙𝑛 2 . 25. 𝑆 𝑦 = 2𝑥 − 𝑥2, 𝑂𝑦, 𝑦 = 1 = |2𝑥 − 𝑥2 − 1| 1 0 𝑑𝑥 = 1 3 . 26. 𝑆 𝑦 = 𝑥4, 𝑦 = 0, 𝑥 = 1 = 𝑥4 1 0 𝑑𝑥 = 1 5 . Dạng 2d. 𝑺 𝒙 = 𝒇 𝒚 , 𝒙 = 𝒈 𝒚 = |𝒇 𝒚 − 𝒈(𝒚)| 𝒃 𝒂 𝒅𝒚. 27. 𝑆 𝑦2 − 2𝑦 + 𝑥 = 0, 𝑥 + 𝑦 = 0 = |𝑦2 − 3𝑦|𝑑𝑦 3 0 = 9 2 . 28. 𝑆 𝑦2 = 2𝑥, 𝑥 − 2𝑦 + 2 = 0, 𝑂𝑥 = ( 𝑦2 2 − 2𝑦 + 2 0 2) 𝑑𝑦 = 4 3 . 29. 𝑆 𝑦 = 𝑥, 𝑦 = 2 − 𝑥, 𝑂𝑥 = |𝑦2 − 2 + 𝑦| 1 0 𝑑𝑦 = 7 6 . 30. 𝑆 𝑦2 = 4𝑥, 𝑦2 = 4 − 𝑥 3 = (4 − 𝑦 2 3 − 2 2 −2 2 1 4 𝑦2) 𝑑𝑦 = 128 2 15 . 31. 𝑆 𝑦2 = 2𝑥, 27𝑦2 = 8 𝑥 − 1 3 = (1 + 3 2 𝑦 2 3 − 2 2 −2 2 1 2 𝑦2) 𝑑𝑦 = 68 2 15 . 32. 𝑆 𝑦 = 𝑥 + 1 2 , 𝑥 = 𝑠𝑖𝑛 𝜋𝑦 , 0 ≤ 𝑦 ≤ 1 = (𝑠𝑖𝑛 𝜋𝑦 − 𝑦 + 1) 1 0 𝑑𝑦 = 2 𝜋 + 1 3 . ---------------------------- Dạng 3. 𝑺{𝒚 = 𝒇 𝒙 , 𝒚 = 𝒈 𝒙 , 𝒚 = 𝒉(𝒙)} 33. A02. S 𝑦 = 𝑥2 − 4𝑥 + 3 , 𝑦 = 𝑥 + 3 = 109 6 . 34. 𝑆 𝑥2 𝑎2 + 𝑦2 𝑏2 = 1 = 4. 𝑏 𝑎 𝑎2 − 𝑥2 𝑎 0 𝑑𝑥 = 𝜋𝑎𝑏. 35. 𝑆 𝑦 = 𝑥2 , 𝑦 = 𝑥2 27 , 𝑦 = 27 𝑥 = (𝑥2 − 𝑥2 27 ) 3 0 𝑑𝑥 + ( 27 𝑥 − 𝑥2 27 ) 9 3 𝑑𝑥 = 27 𝑙𝑛 3. 36. 𝑆 𝑦 = 𝑥2 , 𝑦 = 𝑥2 4 , 𝑦 = 2 𝑥 , 𝑦 = 8 𝑥 = (𝑥2 − 2 𝑥 ) 2 2 3 𝑑𝑥 + ( 8 𝑥 − 𝑥2 4 ) 2 4 3 2 𝑑𝑥 = 4 𝑙𝑛 2. 37. 𝑆 𝑦 = 𝑥2 + 3 2 𝑥 − 3 2 , 𝑦 = 𝑥 = ||𝑑𝑥 = 23 3 . 1 −3 38. 𝑆 𝑥2 = 𝑎𝑦, 𝑦2 = 𝑎𝑥 𝑎 > 0 → 𝑆 = ( 𝑎𝑥 − 𝑎 0 𝑥2 𝑎 ) 𝑑𝑥 = 𝑎2 3 . 39. 𝑆 𝑦 = − 1 3 𝑥2 − 8𝑥 + 7 , 𝑦 = 7−𝑥 𝑥−3 = (− 1 3 𝑥2 − 7 4 8𝑥 + 7 − 7−𝑥 𝑥−3 ) 𝑑𝑥 = 9 + 8 𝑙𝑛 2. 40. 𝑆 𝑦 = 1 − 2 𝑠𝑖𝑛2 3𝑥 2 , 𝑦 = 1 + 12𝑥 𝜋 ; 𝑥 = 𝜋 2 = (1 + 12𝑥 𝜋 − 𝑐𝑜𝑠 3𝑥) 𝜋 6 0 𝑑𝑥 + (1 + 12𝑥 𝜋 + 𝑐𝑜𝑠 3𝑥) 𝜋 2 𝜋 6 𝑑𝑥 = 2𝜋 − 1. 41. 𝑆 𝑦 = 𝑥2 − 2𝑥 + 2, 𝑐á𝑐 𝑡𝑖ế𝑝 𝑡𝑢𝑦ế𝑛 𝑞𝑢𝑎 𝐴 2, −2 . G: = 𝑆 𝑦 = 𝑥2 − 2𝑥 + 2, 𝑦 = −2𝑥 + 2, 𝑦 = 6𝑥 − 14 = 𝑥2 − 2𝑥 + 2 + 2𝑥 − 2 2 0 𝑑𝑥 + 𝑥2 − 2𝑥 + 4 2 2 − 6𝑥 + 14 𝑑𝑥 = 16 3 . 42. Tính tỷ số mà 𝑃 : 𝑦2 = 2𝑥 chia 𝐶 : 𝑥2 + 𝑦2 = 8. HD: 𝑆1 = 2 𝜋 + 2 3 → 𝑆1 𝑆2 = 3𝜋+2 9𝜋−2 . 2. Tính thể tích vật tròn xoay Dạng 1a. 𝑽𝑶𝒙 𝒚 = 𝒇 𝒙 , 𝑶𝒙, 𝒙 = 𝒂, 𝒙 = 𝒃 = 𝝅 𝒇 𝟐(𝒙) 𝒃 𝒂 𝒅𝒙 43. TN07L2b. 𝑉𝑂𝑥 𝑦 = 𝑠𝑖𝑛 𝑥 , 𝑦 = 0, 𝑥 = 0, 𝑥 = 𝜋 2 = 𝜋 𝑠𝑖𝑛2 𝑥 𝑑𝑥 𝜋 2 0 = 𝜋2 4 . 44. B07. 𝑉𝑂𝑥 𝑦 = 𝑥 𝑙𝑛 𝑥 , 𝑦 = 0, 𝑥 = 𝑒 = 𝜋 𝑥 2𝑙𝑛2𝑥 𝑒 1 𝑑𝑥 = 𝜋(5𝑒3−2) 27 . 45. DBA04. 𝑉𝑂𝑥 𝑂𝑥, 𝑦 = 𝑥. 𝑠𝑖𝑛 𝑥 , 0 ≤ 𝑥 ≤ 𝜋 = 𝜋 𝑥 𝜋 0 𝑠𝑖𝑛2𝑥𝑑𝑥 = 𝜋3 4 . 46. 𝑉𝑂𝑥 𝑦 = 𝑡𝑎𝑛 𝑥, 𝑥 = 0, 𝑥 = 𝜋 3 , 𝑦 = 0 = 𝜋 𝑡𝑎𝑛2 𝑥 𝑑𝑥 𝜋 3 0 = 𝜋 3 − 𝜋 3 . 47. 𝑉�