第六章不定积分 1不定积分的概念 求下列不定积分: (3)∫(V+v++)d (4)∫ (5)∫2d (7)/(2sin c-4 cos r)dr (8)∫(3 (9)∫(tan2x+3)dx (10)∫2s2d (11) cos annada (12)∫a2andr; (13)∫rc (14)∫(52+1)2c (15)∫(2+()2-言)dr; (16)∫e2(
第六章 不定积分 §1 不定积分的概念 1.求下列不定积分: (1) R (x 5 + x 3 − √ x 4 )dx; (2) R (5 − x) 3 dx; (3) R ( √ x + √3 x + √ 3 x + 2 √3 x )dx; (4) R dx x4(1+x2); (5) R 3x 2 1+x2 dx; (6) R x√ +1 x dx; (7) R (2 sin x − 4 cos x)dx; (8) R (3 − sec2 x)dx; (9) R (tan2 x + 3)dx; (10) R 2+sin2 x cos2 x dx; (11) R tan x cos2 x 2 −sin2 x 2 dx; (12) R cos 2x cos x−sin x dx; (13) R dx 1+cos 2x; (14) R (5x + 1)2dx; (15) R (2x + ( 1 3 ) x − e x 5 )dx; (16) R e x (1 − e −x √ x )dx; 1
(17)∫(x-12x-ah- (18)∫ dr (19)∫223adx; (20)∫(3+sina)d 2.求一曲线y=f(x),它在点(x,f(x)处的切线的斜率为2x,且通过 点(2,5) 3.已知f(x)满足给定的关系式,试求f(x) (1)rf(x)=1(x>0); ∫(x) (3)f(x)f(x)=1(x>0); 1(f(x)>0) §2换元积分法与分部积分法 用凑微分法求下列不定积分: )∫m2 (3)∫a++a-4; (4)∫(+n3x) (5)∫a+xdx; (6)∫e-dr
(17) R (cos x − 2 1+x2 − 1 4 √ 1−x2 )dx; (18) R px √ xdx; (19) R 2 2x3 xdx; (20) R (√ 3 4−4x2 + sin x)dx. 2.求一曲线y = f(x),它在点(x, f(x))处的切线的斜率为2x ,且通过 点(2, 5). 3.已知f(x)满足给定的关系式,试求f(x): (1) xf0 (x) = 1(x > 0); (2) f 0 (x) x = 1(x > 0); (3) f(x)f 0 (x) = 1(x > 0); (4) f 0 (x) f(x) = 1(f(x) > 0). §2 换元积分法与分部积分法 1.用凑微分法求下列不定积分: (1) R 1 5x−6 dx; (2) R 1 x(1+2x) dx; (3) R √ 1 x+1+√ x−1 dx; (4) R (√ 1 3−x2 + √ 1 1−3x2 )dx; (5) R 1 2+3x2 dx; (6) R e − x 2 dx; 2
(7)re-rd. d 9)∫a+x (10)J=4=; (11)∫ (12)∫ta (13)∫l2s (14)∫ (15) d ∫sg (18)∫ u-do; 19)∫ )∫a )∫ 4)∫sin d 3
(7) R xe−x 2 dx; (8) R e x 1+e x dx; (9) R dx e x+e−x+2; (10) R dx e x+e−x; (11) R tan xdx; (12) R tan5 x sec2 xdx; (13) R 1−2 sin x cos2 x dx; (14) R dx A sin2 x+B cos2 x; (15) R cos5 xdx; (16) R dx 1+sin x; (17) R cos 2x sin x cos x dx; (18) R sin x cos x 1+sin4 x dx; (19) R x 4+x2 dx; (20) R x 4+x4 dx; (21) R 5−4x 3x−2 dx; (22) R sin 2x cos 3xdx; (23) R (ln x) 2 x dx; (24) R sin 1 x . dx x2; (25) R (arcsin x) 2 √ 1−x2 dx; 3
(26)∫+dr; (27)J (28)∫√a (29)J=cdx; (30)J√1+sind 2.用换元积分法求下列不定积分: (2)J=d (3)Jx 2+r-r2d r (5)∫ (6)∫ x+√x2-1 (7)∫em+dx; (8)∫江H+d (10)J+√ (11)J-d; (12)「碳d 3.用分部积分法求下列不定积分:
(26) R arctan x 1+x2 dx; (27) R dx √ x √ 1+√ x ; (28) R √ dx x(1+x); (29) R e x √ 1−e 2x dx; (30) R √ 1 + sin xdx. 2.用换元积分法求下列不定积分: (1) R √ x 2 − a 2dx; (2) R x 2 √ 4−x2 dx; (3) R √ x 5+x−x2 dx; (4) R √ 2 + x − x 2dx; (5) R dx (x2+a 2) 3/2; (6) R dx x+ √ x2−1; (7) R e √ x+1dx; (8) R √ √x+1−1 x+1+1 dx; (9) R √ x 1+√3 x dx; (10) R x 1+√ x dx; (11) R x 5 √ 1−x2 dx; (12) R x 2+2 (x+1)3 dx. 3.用分部积分法求下列不定积分: 4
(1)∫x2 cos ra T (2)「x3lndr; 3)∫lnxd (4)∫ arctan xda (5)∫ (6)∫ r arctan adr; (8)∫cos(lmx)dx d )∫ (12)∫xcos2ada (13)∫[n(x)+mldr (14)∫# (15)∫( arcsin -dr (16)∫=ndr (17)∫ln(x+√1+x2)dr; 18)∫ 4.求下列不定积分的递推公式:
(1) R x 2 cos xdx; (2) R x 3 ln xdx; (3) R ln xdx; (4) R arctan xdx; (5) R arctan √ xdx 1−x ; (6) R x arctan xdx; (7) R ln x x3 dx; (8) R cos(ln x)dx; (9) R sec5 xdx; (10) R x ln( 1+x 1−x )dx; (11) R x sin2 xdx; (12) R x cos2 xdx; (13) R [ln(ln x) + 1 ln x ]dx; (14) R xex (x+1)2 dx; (15) R (arcsin x) 2dx; (16) R x sin2 x dx; (17) R ln(x + √ 1 + x 2)dx; (18) R √ x ln2 xdx. 4.求下列不定积分的递推公式: 5
(1)In=∫r"ekdr; (2)In=∫(nx)dx; (3)In=∫ tanda; (4)In=∫( (arcsin z)dr 5.求下列有理函数的不定积分: (1)「+x (3)∫12xdhr 2 (6)∫ (7)∫xdr; (8)∫=22dr; )Jr+2 0)∫s 6.求下列三角有理式的积分: (2)J5+n2x; (3)J2
(1)In = R x n e kxdx; (2)In = R (ln x) ndx; (3)In = R tann xdx; (4)In = R (arcsin x) ndx. 5.求下列有理函数的不定积分: (1) R x 5+x 4−8 x3−x dx; (2) R dx (x+1)(x2+1)dx; (3) R x 2 1−x4 dx; (4) R dx 1+x3 dx; (5) R x−2 x2−7x+12 dx; (6) R x+4 (x2−1)(x+2)dx; (7) R dx 1+x4 dx; (8) R 2x−3 x2+2x+1 dx; (9) R dx x2+4x+2 dx; (10) R dx 8−2x−x2 dx. 6.求下列三角有理式的积分: (1) R dx 4+5 cos x; (2) R dx 5+4 sin 2x; (3) R dx 2+sin2 x; 6
(4)∫d+x (5) ∫+面m (7)∫ sIn. cosE-dz (8)∫1+2d (9)tanda (10)/anoc (11)∫snx+2 (12) d 7.求下列无理函数的不定积分: (1) x(1+2√x+vx) (2)∫xx+1y=dx (3)∫xm4x (5)∫√H (6)∫ (9)∫ 7
(4) R sec xdx (1+sec x) 2; (5) R dx 1+tan x; (6) R dx (2+cos x) sin x; (7) R sin x cos x sin x+cos x dx; (8) R sin3 x 1+cos2 x dx; (9) R tan3 xdx; (10) R dx sin2 x cos x; (11) R cos xdx sin x+2 cos x dx; (12) R sin2 x 1+cos2 x dx. 7.求下列无理函数的不定积分: (1) R dx x(1+2√ x+ √3 x) dx; (2) R √ x+1− √ √ x−1 x+1+√ x−1 dx; (3) R dx x √ x2+2x+2; (4) R x √ x 2 − 2x + 2dx; (5) R q1−x 1+x dx x2; (6) R √ dx x2+x; (7) R x 2 √ dx 1+x−x2; (8) R √ x 2 + x + 1dx; (9) R dx x √4 1+x4; 7
(10)J√m 8.求下列不定积分: (3)∫ re sin adx; (4)∫ recos rdr; (5)∫ (6)∫ tan a tan(x+a)dx; d 1-+tan (10)∫计adr; (11)∫1tad (12)∫ esn. sin2rdr (13)∫ arctan(1+√a)dr; (14)Ja
(10) R xdx √4 x3(1−x) . 8.求下列不定积分: (1) R √ dx (x−a)(x−b) ; (2) R dx x2 √ αx2+β ; (3) R xex sin xdx; (4) R xex cos xdx; (5) R dx sin(x+a) sin(x+b); (6) R tan x tan(x + α)dx; (7) R x 3 √arccos x 1−x2 dx; (8) R tan x 1+tan x+tan2 x dx; (9) R xex √ dx 1+e x dx; (10) R x+sin x 1+cos x dx; (11) R 1−tan x 1+tan x dx; (12) R e sin x sin 2xdx; (13) R arctan(1 + √ x)dx; (14) R dx (1+2x) 4. 8