T2.1. INDEFINITE INTEGRALS 1137 T2.1.3. Integrals Involving Exponential Functions 1. e ax dx = 1 a e ax . 2. a x dx = a x ln a . 3. xe ax dx = e ax x a – 1 a 2 . 4. x 2 e ax dx = e ax x 2 a – 2x a 2 + 2 a 3 . 5. x n e ax dx = e ax 1 a x n – n a 2 x n–1 + n(n – 1) a 3 x n–2 – ···+(–1) n–1 n! a n x +(–1) n n! a n+1 , n = 1, 2, 6. P n (x)e ax dx = e ax n k=0 (–1) k a k+1 d k dx k P n (x), where P n (x) is an arbitrary polynomial of degree n. 7. dx a + be px = x a – 1 ap ln |a + be px |. 8. dx ae px + be –px = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 p √ ab arctan e px a b if ab > 0, 1 2p √ –ab ln b + e px √ –ab b – e px √ –ab if ab < 0. 9. dx √ a + be px = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 p √ a ln √ a + be px – √ a √ a + be px + √ a if a > 0, 2 p √ –a arctan √ a + be px √ –a if a < 0. T2.1.4. Integrals Involving Hyperbolic Functions T2.1.4-1. Integrals involving cosh x. 1. cosh(a + bx) dx = 1 b sinh(a + bx). 2. x cosh xdx = x sinh x –coshx. 3. x 2 cosh xdx =(x 2 + 2)sinhx – 2x cosh x. 4. x 2n cosh xdx =(2n)! n k=1 x 2k (2k)! sinh x – x 2k–1 (2k – 1)! cosh x . 1138 INTEGRALS 5. x 2n+1 cosh xdx=(2n + 1)! n k=0 x 2k+1 (2k + 1)! sinh x – x 2k (2k)! cosh x . 6. x p cosh xdx= x p sinh x – px p–1 cosh x + p(p – 1) x p–2 cosh xdx. 7. cosh 2 xdx= 1 2 x + 1 4 sinh 2x. 8. cosh 3 xdx=sinhx + 1 3 sinh 3 x. 9. cosh 2n xdx= C n 2n x 2 2n + 1 2 2n–1 n–1 k=0 C k 2n sinh[2(n – k)x] 2(n – k) , n = 1, 2, 10. cosh 2n+1 xdx= 1 2 2n n k=0 C k 2n+1 sinh[(2n – 2k + 1)x] 2n – 2k + 1 = n k=0 C k n sinh 2k+1 x 2k + 1 , n = 1, 2, 11. cosh p xdx= 1 p sinh x cosh p–1 x + p – 1 p cosh p–2 xdx. 12. cosh ax cosh bx dx = 1 a 2 – b 2 (a cosh bx sinh ax – b cosh ax sinh bx). 13. dx cosh ax = 2 a arctan e ax . 14. dx cosh 2n x = sinh x 2n – 1 1 cosh 2n–1 x + n–1 k=1 2 k (n – 1)(n – 2) (n – k) (2n – 3)(2n – 5) (2n – 2k – 1) 1 cosh 2n–2k–1 x , n = 1, 2, 15. dx cosh 2n+1 x = sinh x 2n 1 cosh 2n x + n–1 k=1 (2n – 1)(2n – 3) (2n – 2k + 1) 2 k (n – 1)(n – 2) (n – k) 1 cosh 2n–2k x + (2n – 1)!! (2n)!! arctan sinh x, n = 1, 2, 16. dx a + b cosh x = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ – sign x √ b 2 – a 2 arcsin b + a cosh x a + b cosh x if a 2 < b 2 , 1 √ a 2 – b 2 ln a + b + √ a 2 – b 2 tanh(x/2) a + b – √ a 2 – b 2 tanh(x/2) if a 2 > b 2 . T2.1.4-2. Integrals involving sinh x. 1. sinh(a + bx) dx = 1 b cosh(a + bx). T2.1. INDEFINITE INTEGRALS 1139 2. x sinh xdx= x cosh x –sinhx. 3. x 2 sinh xdx =(x 2 + 2)coshx – 2x sinh x. 4. x 2n sinh xdx =(2n)! n k=0 x 2k (2k)! cosh x – n k=1 x 2k–1 (2k – 1)! sinh x . 5. x 2n+1 sinh xdx=(2n + 1)! n k=0 x 2k+1 (2k + 1)! cosh x – x 2k (2k)! sinh x . 6. x p sinh xdx= x p cosh x – px p–1 sinh x + p(p – 1) x p–2 sinh xdx. 7. sinh 2 xdx=– 1 2 x + 1 4 sinh 2x. 8. sinh 3 xdx=–coshx + 1 3 cosh 3 x. 9. sinh 2n xdx=(–1) n C n 2n x 2 2n + 1 2 2n–1 n–1 k=0 (–1) k C k 2n sinh[2(n – k)x] 2(n – k) , n = 1, 2, 10. sinh 2n+1 xdx= 1 2 2n n k=0 (–1) k C k 2n+1 cosh[(2n – 2k + 1)x] 2n – 2k + 1 = n k=0 (–1) n+k C k n cosh 2k+1 x 2k + 1 , n = 1, 2, 11. sinh p xdx= 1 p sinh p–1 x cosh x – p – 1 p sinh p–2 xdx. 12. sinh ax sinh bx dx = 1 a 2 – b 2 a cosh ax sinh bx – b cosh bx sinh ax . 13. dx sinh ax = 1 a ln tanh ax 2 . 14. dx sinh 2n x = cosh x 2n – 1 – 1 sinh 2n–1 x + n–1 k=1 (–1) k–1 2 k (n – 1)(n – 2) (n – k) (2n – 3)(2n – 5) (2n – 2k – 1) 1 sinh 2n–2k–1 x , n = 1, 2, 15. dx sinh 2n+1 x = cosh x 2n – 1 sinh 2n x + n–1 k=1 (–1) k–1 (2n–1)(2n–3) (2n–2k+1) 2 k (n–1)(n–2) (n–k) 1 sinh 2n–2k x +(–1) n (2n–1)!! (2n)!! ln tanh x 2 , n = 1, 2, 16. dx a + b sinh x = 1 √ a 2 + b 2 ln a tanh(x/2)–b + √ a 2 + b 2 a tanh(x/2)–b – √ a 2 + b 2 . 17. Ax + B sinh x a + b sinh x dx = B b x + Ab – Ba b √ a 2 + b 2 ln a tanh(x/2)–b + √ a 2 + b 2 a tanh(x/2)–b – √ a 2 + b 2 . 1140 INTEGRALS T2.1.4-3. Integrals involving tanh x or cothx. 1. tanh xdx=lncoshx. 2. tanh 2 xdx= x –tanhx. 3. tanh 3 xdx=– 1 2 tanh 2 x +lncoshx. 4. tanh 2n xdx= x – n k=1 tanh 2n–2k+1 x 2n – 2k + 1 , n = 1, 2, 5. tanh 2n+1 xdx=lncoshx – n k=1 (–1) k C k n 2k cosh 2k x =lncoshx – n k=1 tanh 2n–2k+2 x 2n – 2k + 2 , n = 1, 2, 6. tanh p xdx=– 1 p – 1 tanh p–1 x + tanh p–2 xdx. 7. coth xdx =ln|sinh x|. 8. coth 2 xdx= x –cothx. 9. coth 3 xdx=– 1 2 coth 2 x +ln|sinh x|. 10. coth 2n xdx= x – n k=1 coth 2n–2k+1 x 2n – 2k + 1 , n = 1, 2, 11. coth 2n+1 xdx=ln|sinh x| – n k=1 C k n 2k sinh 2k x =ln|sinh x| – n k=1 coth 2n–2k+2 x 2n – 2k + 2 , n = 1, 2, 12. coth p xdx=– 1 p – 1 coth p–1 x + coth p–2 xdx. T2.1.5. Integrals Involving Logarithmic Functions 1. ln ax dx = x ln ax – x. 2. x ln xdx = 1 2 x 2 ln x – 1 4 x 2 . 3. x p ln ax dx = ⎧ ⎨ ⎩ 1 p + 1 x p+1 ln ax – 1 (p + 1) 2 x p+1 if p ≠ –1, 1 2 ln 2 ax if p =–1. T2.1. INDEFINITE INTEGRALS 1141 4. (ln x) 2 dx = x(ln x) 2 – 2x ln x + 2x. 5. x(ln x) 2 dx = 1 2 x 2 (ln x) 2 – 1 2 x 2 ln x + 1 4 x 2 . 6. x p (ln x) 2 dx = ⎧ ⎪ ⎨ ⎪ ⎩ x p+1 p + 1 (ln x) 2 – 2x p+1 (p + 1) 2 ln x + 2x p+1 (p + 1) 3 if p ≠ –1, 1 3 ln 3 x if p =–1. 7. (ln x) n dx = x n + 1 n k=0 (–1) k (n + 1)n (n – k + 1)(ln x) n–k , n = 1, 2, 8. (ln x) q dx = x(ln x) q – q (ln x) q–1 dx, q ≠ –1. 9. x n (ln x) m dx = x n+1 m + 1 m k=0 (–1) k (n + 1) k+1 (m + 1)m (m – k + 1)(ln x) m–k , n, m = 1, 2, 10. x p (ln x) q dx = 1 p + 1 x p+1 (ln x) q – q p + 1 x p (ln x) q–1 dx, p, q ≠ –1. 11. ln(a + bx) dx = 1 b (ax + b)ln(ax + b)–x. 12. x ln(a + bx) dx = 1 2 x 2 – a 2 b 2 ln(a + bx)– 1 2 x 2 2 – a b x . 13. x 2 ln(a + bx) dx = 1 3 x 3 – a 3 b 3 ln(a + bx)– 1 3 x 3 3 – ax 2 2b + a 2 x b 2 . 14. ln xdx (a + bx) 2 =– ln x b(a + bx) + 1 ab ln x a + bx . 15. ln xdx (a + bx) 3 =– ln x 2b(a + bx) 2 + 1 2ab(a + bx) + 1 2a 2 b ln x a + bx . 16. ln xdx √ a + bx = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 b (ln x – 2) √ a + bx + √ a ln √ a + bx + √ a √ a + bx – √ a if a > 0, 2 b (ln x – 2) √ a + bx + 2 √ –a arctan √ a + bx √ –a if a < 0. 17. ln(x 2 + a 2 ) dx = x ln(x 2 + a 2 )–2x + 2a arctan(x/a). 18. x ln(x 2 + a 2 ) dx = 1 2 (x 2 + a 2 )ln(x 2 + a 2 )–x 2 . 19. x 2 ln(x 2 + a 2 ) dx = 1 3 x 3 ln(x 2 + a 2 )– 2 3 x 3 + 2a 2 x – 2a 3 arctan(x/a) . 1142 INTEGRALS T2.1.6. Integrals Involving Trigonometric Functions T2.1.6-1. Integrals involving cos x (n = 1, 2, ). 1. cos(a + bx) dx = 1 b sin(a + bx). 2. x cos xdx =cosx + x sin x. 3. x 2 cos xdx= 2x cos x +(x 2 – 2)sinx. 4. x 2n cos xdx=(2n)! n k=0 (–1) k x 2n–2k (2n – 2k)! sin x + n–1 k=0 (–1) k x 2n–2k–1 (2n – 2k – 1)! cos x . 5. x 2n+1 cos xdx =(2n + 1)! n k=0 (–1) k x 2n–2k+1 (2n – 2k + 1)! sin x + x 2n–2k (2n – 2k)! cos x . 6. x p cos xdx = x p sin x + px p–1 cos x – p(p – 1) x p–2 cos xdx. 7. cos 2 xdx= 1 2 x + 1 4 sin 2x. 8. cos 3 xdx=sinx – 1 3 sin 3 x. 9. cos 2n xdx= 1 2 2n C n 2n x + 1 2 2n–1 n–1 k=0 C k 2n sin[(2n – 2k)x] 2n – 2k . 10. cos 2n+1 xdx= 1 2 2n n k=0 C k 2n+1 sin[(2n – 2k + 1)x] 2n – 2k + 1 . 11. dx cos x =ln tan x 2 + π 4 . 12. dx cos 2 x =tanx. 13. dx cos 3 x = sin x 2 cos 2 x + 1 2 ln tan x 2 + π 4 . 14. dx cos n x = sin x (n – 1)cos n–1 x + n – 2 n – 1 dx cos n–2 x , n > 1. 15. xdx cos 2n x = n–1 k=0 (2n–2)(2n–4) (2n–2k +2) (2n–1)(2n–3) (2n–2k +3) (2n–2k)x sin x–cosx (2n–2k +1)(2n–2k)cos 2n–2k+1 x + 2 n–1 (n–1)! (2n–1)!! x tan x +ln|cos x| . 16. cos ax cosbx dx = sin (b – a)x 2(b – a) + sin (b + a)x 2(b + a) , a ≠ b. T2.1. INDEFINITE INTEGRALS 1143 17. dx a + b cos x = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 √ a 2 – b 2 arctan (a – b)tan(x/2) √ a 2 – b 2 if a 2 > b 2 , 1 √ b 2 – a 2 ln √ b 2 – a 2 +(b – a)tan(x/2) √ b 2 – a 2 –(b – a)tan(x/2) if b 2 > a 2 . 18. dx (a + b cos x) 2 = b sin x (b 2 – a 2 )(a + b cos x) – a b 2 – a 2 dx a + b cos x . 19. dx a 2 + b 2 cos 2 x = 1 a √ a 2 + b 2 arctan a tan x √ a 2 + b 2 . 20. dx a 2 – b 2 cos 2 x = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 a √ a 2 – b 2 arctan a tan x √ a 2 – b 2 if a 2 > b 2 , 1 2a √ b 2 – a 2 ln √ b 2 – a 2 – a tan x √ b 2 – a 2 + a tan x if b 2 > a 2 . 21. e ax cos bx dx = e ax b a 2 + b 2 sin bx + a a 2 + b 2 cos bx . 22. e ax cos 2 xdx= e ax a 2 + 4 a cos 2 x + 2 sin x cos x + 2 a . 23. e ax cos n xdx= e ax cos n–1 x a 2 + n 2 (a cos x + n sin x)+ n(n – 1) a 2 + n 2 e ax cos n–2 xdx. T2.1.6-2. Integrals involving sin x (n = 1, 2, ). 1. sin(a + bx) dx =– 1 b cos(a + bx). 2. x sin xdx =sinx – x cos x. 3. x 2 sin xdx= 2x sin x –(x 2 – 2)cosx. 4. x 3 sin xdx=(3x 2 – 6)sinx –(x 3 – 6x)cosx. 5. x 2n sin xdx=(2n)! n k=0 (–1) k+1 x 2n–2k (2n – 2k)! cos x + n–1 k=0 (–1) k x 2n–2k–1 (2n – 2k – 1)! sin x . 6. x 2n+1 sin xdx=(2n+1)! n k=0 (–1) k+1 x 2n–2k+1 (2n – 2k + 1)! cos x+(–1) k x 2n–2k (2n – 2k)! sin x . 7. x p sin xdx =–x p cos x + px p–1 sin x – p(p – 1) x p–2 sin xdx. 8. sin 2 xdx= 1 2 x – 1 4 sin 2x. . 6. P n (x)e ax dx = e ax n k=0 (–1) k a k+1 d k dx k P n (x), where P n (x) is an arbitrary polynomial of degree n. 7. dx a + be px = x a – 1 ap ln |a + be px |. 8. dx ae px + be –px = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 p √ ab arctan e px a b if