Chuyen de Tich Phan
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<span class='text_page_counter'>(1)</span>CHUYÊN ĐỀ: NGUYÊN HÀM - TÍCH PHÂN I. LÝ THUYẾT 1) Bảng nguyên hàm các hàm số thường gặp. F’(x) 0 x. α. 1 x e. x. a. x. Nguyên hàm C x ∝+1 + C , ∝≠−1 ∝+1 ln |x|+C x. e +C x. a +C ln a. 1 x2. −1 +C x. cos x sin x 1 cos2 x 1 sin2 x. sin x+C −cos x +C. u. ∝. 1 u e. u. a. u. 1 u2 cos u sin u. tan x +C −cot x+ C. u∝ +C u ' (∝+ 1) ln|u| +C u' eu +C u' au +C u ' (ln a) −1 +C u' . u sinu +C u' −cos u +C u'. Lưu ý: Hàm số u thường là u=f(x)=ax+b và u’=a 2) Phương pháp đổi biến số Đặt u=f(x) u’.du=f’(x).dx Đưa tích phân ban đầu về một tích phân mới với biến u. Hệ quả:. 1. ∫ f ( ax+ b ) dx= a F ( ax+ b ) +C , với a ≠ 0. Lưu ý: Đặt u thường là các hàm có dấu căn, lượng giác, lnx, hàm mũ,...
<span class='text_page_counter'>(2)</span> 3) Phương pháp nguyên hàm / tích phân từng phần a. b. ∫ udv=uv ∨. ∫ udv=uv −∫ vdu. a. b. b−∫ vdu a. Lưu ý: a. Nhận biết: khi đặt ẩn ta không thể biến đồi các biểu thức về cùng ẩn vừa đặt b. Thường ưu tiên đặt u là các hàm lnx, hàm đa thức (x, 1-x, 4x+3,..) c. Thường ưu tiên đặt dv là các hàm sin, cos, ex, hàm mũ… d. Đối với các hàm lượng giác có số mũ từ 2 trở lên thì dùng Công thức hạ bậc: 1−cos 2 x 1+cos 2 x ; cos2 x= 2 2 3 sinx−sin 3 x cos 3 x +3 cosx 3 3 sin x= ; cos x = 4 4 sin 2 x=. 4) Các tính chất của tích phân. b. a ¿ ∫ f ( x ) dx=F ( x )∨b =F ( b )−F (a) a a b. a. b ¿∫ f ( x ) dx=−∫ f ( x ) dx a c. b. b. b. c ¿∫ f ( x ) dx +∫ f ( x ) dx=∫ f ( x ) dx , với a< c<b a. c. a. II. BÀI TẬP Bài 1. Tính các nguyên hàm sau: 1¿ ∫ ( 3 x +e 1−2 x ) dx 1 2 ¿∫ dx (1+ x )(1−2 x) 3 ¿∫ sin3 x cos 5 x dx 4 ¿∫ ( 1−x )4 dx 3. 5 ¿∫ x 2 √1+ x 3 dx với x>−1 2. 6 ¿∫ x e− x dx x 7 ¿∫ dx 2 ( 1+ x 2 ) 1 8 ¿∫ dx (1−x ) √ x ( lnx )2 9 ¿∫ dx x sinx 10 ¿∫ 3 dx √ cos 2 x 1 11 ¿∫ x −x dx e −e 12¿ ∫ x ( 3−x )5 dx 13 ¿∫ x √ 2−5 x dx 14 ¿ ∫ xln ( 1+ x ) dx 15 ¿∫ (1−2 x ) e x dx 16 ¿∫ ( x 2+2 x−1 ) e x dx. 17 ¿∫ ( 1−x ) cosxdx 18 ¿∫ xsi n2 xdx 2. 19 ¿∫ √ x ln x dx x−1 20 ¿∫ xln dx x+1 x cos ¿ ¿ ¿ 2 cos x ln ¿ ¿ 21 ¿∫ ¿ x 22 ¿∫ 2 dx sin x 23 ¿∫ ( x+lnx ) x 2 dx 24 ¿ ∫ ( x +e x ) e 2 x dx x 2 x+ sin ¿ sin xdx ¿ ¿ 25 ¿ ∫ ¿.
<span class='text_page_counter'>(3)</span> e x cos x + ( e x +1 ) sin x 26 ¿∫ dx e x sin x x dx x+ sin ¿ cos 2 x ¿ ¿ 27 ¿∫ ¿. 28 ¿∫ (2 x−3) √ x−3 dx. Bài 2. Tính các tích phân sau: 2. 1 1¿ ∫ 2 dx 0 x −2 x −3 2. 2 ¿∫ x ( x+1 )2 dx 0 2. 3 ¿∫ 1 π 2. 1. 17 ¿∫ ln ( 1+ x ) dx 0 π 2. 18 ¿∫ ( x +1 ) sin x dx 0 e. 1−3 x dx ( x +1 )2. 19 ¿∫ x 2 ln x dx 1 ln 2. 4 ¿ ∫ sin x cos 5 x dx −π 2 2. 5 ¿∫|1−x|dx 0 π 2. 2. 6 ¿∫ sin x dx 0 ln 2. e2 x +1+1 dx ex 0 3 x2 8 ¿∫ dx 3 0 2 ( 1+ x ) 7¿∫. 1. 9 ¿∫ √ 1−x 2 dx 0. 1. 1 dx 2 0 1+ x 1 e x (1+ x ) 11 ¿∫ dx x 0 1+ x e 10 ¿∫. a 2. 1 dx , với a >0 2 0 √ a −x a 1 13 ¿∫ 2 2 dx , với a>0 0 a +x 12¿ ∫. 9. 2. 3. 14 ¿ ∫ x √ 1−x dx 1 4. 15 ¿∫ 0 1. 4 x−1 dx √ 2 x +1+2 4. 16 ¿ ∫ x 2 ( 1−x3 ) dx −1. 20 ¿ ∫ x e−2 x dx 0.
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<span class='text_page_counter'>(5)</span> 7. e3. dx 36 ¿∫ 3 1 x √ 1+3 lnx π 2. 37 ¿∫ π 3. sin x dx cos 2 x−cos x. 0. 2 √2. 3+ ln x dx ( x +1 )2. e. 40 ¿ ∫ ¿. dx x √ x 2+ 1 2. 43 ¿ ∫. 1−2 sin x dx 1+sin 2 x. 44 ¿∫. x dx 1+ √ 1−x. 0 2. x 2+ ln ¿ ¿ ¿2 x¿ ln x ¿ 1. 41 ¿∫ ¿. π 4. 38 ¿∫ ¿. 1. π 2. √3. π 2. 39 ¿∫. 2 x +¿ sin x dx √1+3 cos x. 42 ¿ ∫. x−1 3 cos ¿ ¿ ¿. 0 3. sin. 1 3. 45 ¿ ∫ ln ( x 2−x ) dx 2.
<span class='text_page_counter'>(6)</span> KIỂM TRA 15 PHÚT MÔN: GIẢI TÍCH 12 - CHƯƠNG 3 ĐỀ: Bài 1: (7,0 điểm) Tìm các nguyên hàm sau: x 2−x +cos ¿ ¿ a ¿∫ x2 ¿ 1 b ¿∫ (x + ) ln 2 x dx x. Bài 2: (3,0 điểm) Tìm hàm số F(x) biết rằng F(x) là một nguyên hàm của f(x)=3x2-2 và F(-1)=5. Bài làm .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .....................................................................................................................................................
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<span class='text_page_counter'>(17)</span> KIỂM TRA 15 PHÚT MÔN: GIẢI TÍCH 12 - CHƯƠNG 3 ĐỀ: Tính các tích phân sau: π 2. 1¿ ∫ sin 2 x √ 1+3 cos x dx(3,0 đ ) 0 ln3. 2 ¿ ∫ ( 2 x−e2 x ) e x dx (4,0 đ ) 0 2. 1 1 1 3 ¿∫ (1+2 x+ )(2− 2 )ln 2 x+ dx (3,0 đ ) x x x 1. (. ). 2. Bài làm .....................................................................................................................................................
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