1130 INTEGRALS 2. dx (a + x)(b + x) = 1 a – b ln b + x a + x , a ≠ b.Fora = b, see Integral 2 with n =–2 in Paragraph T2.1.1-1. 3. xdx (a + x)(b + x) = 1 a – b a ln |a + x| – b ln |b + x| . 4. dx (a + x)(b + x) 2 = 1 (b – a)(b + x) + 1 (a – b) 2 ln a + x b + x . 5. xdx (a + x)(b + x) 2 = b (a – b)(b + x) – a (a – b) 2 ln a + x b + x . 6. x 2 dx (a + x)(b + x) 2 = b 2 (b – a)(b + x) + a 2 (a – b) 2 ln |a + x| + b 2 – 2ab (b – a) 2 ln |b + x|. 7. dx (a + x) 2 (b + x) 2 =– 1 (a – b) 2 1 a + x + 1 b + x + 2 (a – b) 3 ln a + x b + x . 8. xdx (a + x) 2 (b + x) 2 = 1 (a – b) 2 a a + x + b b + x + a + b (a – b) 3 ln a + x b + x . 9. x 2 dx (a + x) 2 (b + x) 2 =– 1 (a – b) 2 a 2 a + x + b 2 b + x + 2ab (a – b) 3 ln a + x b + x . T2.1.1-3. Integrals involving a 2 + x 2 . 1. dx a 2 + x 2 = 1 a arctan x a . 2. dx (a 2 + x 2 ) 2 = x 2a 2 (a 2 + x 2 ) + 1 2a 3 arctan x a . 3. dx (a 2 + x 2 ) 3 = x 4a 2 (a 2 + x 2 ) 2 + 3x 8a 4 (a 2 + x 2 ) + 3 8a 5 arctan x a . 4. dx (a 2 + x 2 ) n+1 = x 2na 2 (a 2 + x 2 ) n + 2n – 1 2na 2 dx (a 2 + x 2 ) n ; n = 1, 2, 5. xdx a 2 + x 2 = 1 2 ln(a 2 + x 2 ). 6. xdx (a 2 + x 2 ) 2 =– 1 2(a 2 + x 2 ) . 7. xdx (a 2 + x 2 ) 3 =– 1 4(a 2 + x 2 ) 2 . 8. xdx (a 2 + x 2 ) n+1 =– 1 2n(a 2 + x 2 ) n ; n = 1, 2, 9. x 2 dx a 2 + x 2 = x – a arctan x a . 10. x 2 dx (a 2 + x 2 ) 2 =– x 2(a 2 + x 2 ) + 1 2a arctan x a . T2.1. INDEFINITE INTEGRALS 1131 11. x 2 dx (a 2 + x 2 ) 3 =– x 4(a 2 + x 2 ) 2 + x 8a 2 (a 2 + x 2 ) + 1 8a 3 arctan x a . 12. x 2 dx (a 2 + x 2 ) n+1 =– x 2n(a 2 + x 2 ) n + 1 2n dx (a 2 + x 2 ) n ; n = 1, 2, 13. x 3 dx a 2 + x 2 = x 2 2 – a 2 2 ln(a 2 + x 2 ). 14. x 3 dx (a 2 + x 2 ) 2 = a 2 2(a 2 + x 2 ) + 1 2 ln(a 2 + x 2 ). 15. x 3 dx (a 2 + x 2 ) n+1 =– 1 2(n – 1)(a 2 + x 2 ) n–1 + a 2 2n(a 2 + x 2 ) n ; n = 2, 3, 16. dx x(a 2 + x 2 ) = 1 2a 2 ln x 2 a 2 + x 2 . 17. dx x(a 2 + x 2 ) 2 = 1 2a 2 (a 2 + x 2 ) + 1 2a 4 ln x 2 a 2 + x 2 . 18. dx x(a 2 + x 2 ) 3 = 1 4a 2 (a 2 + x 2 ) 2 + 1 2a 4 (a 2 + x 2 ) + 1 2a 6 ln x 2 a 2 + x 2 . 19. dx x 2 (a 2 + x 2 ) =– 1 a 2 x – 1 a 3 arctan x a . 20. dx x 2 (a 2 + x 2 ) 2 =– 1 a 4 x – x 2a 4 (a 2 + x 2 ) – 3 2a 5 arctan x a . 21. dx x 3 (a 2 + x 2 ) 2 =– 1 2a 4 x 2 – 1 2a 4 (a 2 + x 2 ) – 1 a 6 ln x 2 a 2 + x 2 . 22. dx x 2 (a 2 + x 2 ) 3 =– 1 a 6 x – x 4a 4 (a 2 + x 2 ) 2 – 7x 8a 6 (a 2 + x 2 ) – 15 8a 7 arctan x a . 23. dx x 3 (a 2 + x 2 ) 3 =– 1 2a 6 x 2 – 1 a 6 (a 2 + x 2 ) – 1 4a 4 (a 2 + x 2 ) 2 – 3 2a 8 ln x 2 a 2 + x 2 . T2.1.1-4. Integrals involving a 2 – x 2 . 1. dx a 2 – x 2 = 1 2a ln a + x a – x . 2. dx (a 2 – x 2 ) 2 = x 2a 2 (a 2 – x 2 ) + 1 4a 3 ln a + x a – x . 3. dx (a 2 – x 2 ) 3 = x 4a 2 (a 2 – x 2 ) 2 + 3x 8a 4 (a 2 – x 2 ) + 3 16a 5 ln a + x a – x . 4. dx (a 2 – x 2 ) n+1 = x 2na 2 (a 2 – x 2 ) n + 2n – 1 2na 2 dx (a 2 – x 2 ) n ; n = 1, 2, 5. xdx a 2 – x 2 =– 1 2 ln |a 2 – x 2 |. 1132 INTEGRALS 6. xdx (a 2 – x 2 ) 2 = 1 2(a 2 – x 2 ) . 7. xdx (a 2 – x 2 ) 3 = 1 4(a 2 – x 2 ) 2 . 8. xdx (a 2 – x 2 ) n+1 = 1 2n(a 2 – x 2 ) n ; n = 1, 2, 9. x 2 dx a 2 – x 2 =–x + a 2 ln a + x a – x . 10. x 2 dx (a 2 – x 2 ) 2 = x 2(a 2 – x 2 ) – 1 4a ln a + x a – x . 11. x 2 dx (a 2 – x 2 ) 3 = x 4(a 2 – x 2 ) 2 – x 8a 2 (a 2 – x 2 ) – 1 16a 3 ln a + x a – x . 12. x 2 dx (a 2 – x 2 ) n+1 = x 2n(a 2 – x 2 ) n – 1 2n dx (a 2 – x 2 ) n ; n = 1, 2, 13. x 3 dx a 2 – x 2 =– x 2 2 – a 2 2 ln |a 2 – x 2 |. 14. x 3 dx (a 2 – x 2 ) 2 = a 2 2(a 2 – x 2 ) + 1 2 ln |a 2 – x 2 |. 15. x 3 dx (a 2 – x 2 ) n+1 =– 1 2(n – 1)(a 2 – x 2 ) n–1 + a 2 2n(a 2 – x 2 ) n ; n = 2, 3, 16. dx x(a 2 – x 2 ) = 1 2a 2 ln x 2 a 2 – x 2 . 17. dx x(a 2 – x 2 ) 2 = 1 2a 2 (a 2 – x 2 ) + 1 2a 4 ln x 2 a 2 – x 2 . 18. dx x(a 2 – x 2 ) 3 = 1 4a 2 (a 2 – x 2 ) 2 + 1 2a 4 (a 2 – x 2 ) + 1 2a 6 ln x 2 a 2 – x 2 . T2.1.1-5. Integrals involving a 3 + x 3 . 1. dx a 3 + x 3 = 1 6a 2 ln (a + x) 2 a 2 – ax + x 2 + 1 a 2 √ 3 arctan 2x – a a √ 3 . 2. dx (a 3 + x 3 ) 2 = x 3a 3 (a 3 + x 3 ) + 2 3a 3 dx a 3 + x 3 . 3. xdx a 3 + x 3 = 1 6a ln a 2 – ax + x 2 (a + x) 2 + 1 a √ 3 arctan 2x – a a √ 3 . 4. xdx (a 3 + x 3 ) 2 = x 2 3a 3 (a 3 + x 3 ) + 1 3a 3 xdx a 3 + x 3 . 5. x 2 dx a 3 + x 3 = 1 3 ln |a 3 + x 3 |. T2.1. INDEFINITE INTEGRALS 1133 6. dx x(a 3 + x 3 ) = 1 3a 3 ln x 3 a 3 + x 3 . 7. dx x(a 3 + x 3 ) 2 = 1 3a 3 (a 3 + x 3 ) + 1 3a 6 ln x 3 a 3 + x 3 . 8. dx x 2 (a 3 + x 3 ) =– 1 a 3 x – 1 a 3 xdx a 3 + x 3 . 9. dx x 2 (a 3 + x 3 ) 2 =– 1 a 6 x – x 2 3a 6 (a 3 + x 3 ) – 4 3a 6 xdx a 3 + x 3 . T2.1.1-6. Integrals involving a 3 – x 3 . 1. dx a 3 – x 3 = 1 6a 2 ln a 2 + ax + x 2 (a – x) 2 + 1 a 2 √ 3 arctan 2x + a a √ 3 . 2. dx (a 3 – x 3 ) 2 = x 3a 3 (a 3 – x 3 ) + 2 3a 3 dx a 3 – x 3 . 3. xdx a 3 – x 3 = 1 6a ln a 2 + ax + x 2 (a – x) 2 – 1 a √ 3 arctan 2x + a a √ 3 . 4. xdx (a 3 – x 3 ) 2 = x 2 3a 3 (a 3 – x 3 ) + 1 3a 3 xdx a 3 – x 3 . 5. x 2 dx a 3 – x 3 =– 1 3 ln |a 3 – x 3 |. 6. dx x(a 3 – x 3 ) = 1 3a 3 ln x 3 a 3 – x 3 . 7. dx x(a 3 – x 3 ) 2 = 1 3a 3 (a 3 – x 3 ) + 1 3a 6 ln x 3 a 3 – x 3 . 8. dx x 2 (a 3 – x 3 ) =– 1 a 3 x + 1 a 3 xdx a 3 – x 3 . 9. dx x 2 (a 3 – x 3 ) 2 =– 1 a 6 x – x 2 3a 6 (a 3 – x 3 ) + 4 3a 6 xdx a 3 – x 3 . T2.1.1-7. Integrals involving a 4 x 4 . 1. dx a 4 + x 4 = 1 4a 3 √ 2 ln a 2 + ax √ 2 + x 2 a 2 – ax √ 2 + x 2 + 1 2a 3 √ 2 arctan ax √ 2 a 2 – x 2 . 2. xdx a 4 + x 4 = 1 2a 2 arctan x 2 a 2 . 3. x 2 dx a 4 + x 4 =– 1 4a √ 2 ln a 2 + ax √ 2 + x 2 a 2 – ax √ 2 + x 2 + 1 2a √ 2 arctan ax √ 2 a 2 – x 2 . 1134 INTEGRALS 4. dx a 4 – x 4 = 1 4a 3 ln a + x a – x + 1 2a 3 arctan x a . 5. xdx a 4 – x 4 = 1 4a 2 ln a 2 + x 2 a 2 – x 2 . 6. x 2 dx a 4 – x 4 = 1 4a ln a + x a – x – 1 2a arctan x a . T2.1.2. Integrals Involving Irrational Functions T2.1.2-1. Integrals involving x 1/2 . 1. x 1/2 dx a 2 + b 2 x = 2 b 2 x 1/2 – 2a b 3 arctan bx 1/2 a . 2. x 3/2 dx a 2 + b 2 x = 2x 3/2 3b 2 – 2a 2 x 1/2 b 4 + 2a 3 b 5 arctan bx 1/2 a . 3. x 1/2 dx (a 2 + b 2 x) 2 =– x 1/2 b 2 (a 2 + b 2 x) + 1 ab 3 arctan bx 1/2 a . 4. x 3/2 dx (a 2 + b 2 x) 2 = 2x 3/2 b 2 (a 2 + b 2 x) + 3a 2 x 1/2 b 4 (a 2 + b 2 x) – 3a b 5 arctan bx 1/2 a . 5. dx (a 2 + b 2 x)x 1/2 = 2 ab arctan bx 1/2 a . 6. dx (a 2 + b 2 x)x 3/2 =– 2 a 2 x 1/2 – 2b a 3 arctan bx 1/2 a . 7. dx (a 2 + b 2 x) 2 x 1/2 = x 1/2 a 2 (a 2 + b 2 x) + 1 a 3 b arctan bx 1/2 a . 8. x 1/2 dx a 2 – b 2 x =– 2 b 2 x 1/2 + 2a b 3 ln a + bx 1/2 a – bx 1/2 . 9. x 3/2 dx a 2 – b 2 x =– 2x 3/2 3b 2 – 2a 2 x 1/2 b 4 + a 3 b 5 ln a + bx 1/2 a – bx 1/2 . 10. x 1/2 dx (a 2 – b 2 x) 2 = x 1/2 b 2 (a 2 – b 2 x) – 1 2ab 3 ln a + bx 1/2 a – bx 1/2 . 11. x 3/2 dx (a 2 – b 2 x) 2 = 3a 2 x 1/2 – 2b 2 x 3/2 b 4 (a 2 – b 2 x) – 3a 2b 5 ln a + bx 1/2 a – bx 1/2 . 12. dx (a 2 – b 2 x)x 1/2 = 1 ab ln a + bx 1/2 a – bx 1/2 . 13. dx (a 2 – b 2 x)x 3/2 =– 2 a 2 x 1/2 + b a 3 ln a + bx 1/2 a – bx 1/2 . 14. dx (a 2 – b 2 x) 2 x 1/2 = x 1/2 a 2 (a 2 – b 2 x) + 1 2a 3 b ln a + bx 1/2 a – bx 1/2 . T2.1. INDEFINITE INTEGRALS 1135 T2.1.2-2. Integrals involving (a + bx) p/2 . 1. (a + bx) p/2 dx = 2 b(p + 2) (a + bx) (p+2)/2 . 2. x(a + bx) p/2 dx = 2 b 2 (a + bx) (p+4)/2 p + 4 – a(a + bx) (p+2)/2 p + 2 . 3. x 2 (a + bx) p/2 dx = 2 b 3 (a + bx) (p+6)/2 p + 6 – 2a(a + bx) (p+4)/2 p + 4 + a 2 (a + bx) (p+2)/2 p + 2 . T2.1.2-3. Integrals involving (x 2 + a 2 ) 1/2 . 1. (x 2 + a 2 ) 1/2 dx = 1 2 x(a 2 + x 2 ) 1/2 + a 2 2 ln x +(x 2 + a 2 ) 1/2 . 2. x(x 2 + a 2 ) 1/2 dx = 1 3 (a 2 + x 2 ) 3/2 . 3. (x 2 + a 2 ) 3/2 dx = 1 4 x(a 2 + x 2 ) 3/2 + 3 8 a 2 x(a 2 + x 2 ) 1/2 + 3 8 a 4 ln x +(x 2 + a 2 ) 1/2 . 4. 1 x (x 2 + a 2 ) 1/2 dx =(a 2 + x 2 ) 1/2 – a ln a +(x 2 + a 2 ) 1/2 x . 5. dx √ x 2 + a 2 =ln x +(x 2 + a 2 ) 1/2 . 6. xdx √ x 2 + a 2 =(x 2 + a 2 ) 1/2 . 7. (x 2 + a 2 ) –3/2 dx = a –2 x(x 2 + a 2 ) –1/2 . T2.1.2-4. Integrals involving (x 2 – a 2 ) 1/2 . 1. (x 2 – a 2 ) 1/2 dx = 1 2 x(x 2 – a 2 ) 1/2 – a 2 2 ln x +(x 2 – a 2 ) 1/2 . 2. x(x 2 – a 2 ) 1/2 dx = 1 3 (x 2 – a 2 ) 3/2 . 3. (x 2 – a 2 ) 3/2 dx = 1 4 x(x 2 – a 2 ) 3/2 – 3 8 a 2 x(x 2 – a 2 ) 1/2 + 3 8 a 4 ln x +(x 2 – a 2 ) 1/2 . 4. 1 x (x 2 – a 2 ) 1/2 dx =(x 2 – a 2 ) 1/2 – a arccos a x . 5. dx √ x 2 – a 2 =ln x +(x 2 – a 2 ) 1/2 . 6. xdx √ x 2 – a 2 =(x 2 – a 2 ) 1/2 . 7. (x 2 – a 2 ) –3/2 dx =–a –2 x(x 2 – a 2 ) –1/2 . 1136 INTEGRALS T2.1.2-5. Integrals involving (a 2 – x 2 ) 1/2 . 1. (a 2 – x 2 ) 1/2 dx = 1 2 x(a 2 – x 2 ) 1/2 + a 2 2 arcsin x a . 2. x(a 2 – x 2 ) 1/2 dx =– 1 3 (a 2 – x 2 ) 3/2 . 3. (a 2 – x 2 ) 3/2 dx = 1 4 x(a 2 – x 2 ) 3/2 + 3 8 a 2 x(a 2 – x 2 ) 1/2 + 3 8 a 4 arcsin x a . 4. 1 x (a 2 – x 2 ) 1/2 dx =(a 2 – x 2 ) 1/2 – a ln a +(a 2 – x 2 ) 1/2 x . 5. dx √ a 2 – x 2 =arcsin x a . 6. xdx √ a 2 – x 2 =–(a 2 – x 2 ) 1/2 . 7. (a 2 – x 2 ) –3/2 dx = a –2 x(a 2 – x 2 ) –1/2 . T2.1.2-6. Integrals involving arbitrary powers. Reduction formulas. 1. dx x(ax n + b) = 1 bn ln x n ax n + b . 2. dx x √ x n + a 2 = 2 an ln x n/2 √ x n + a 2 + a . 3. dx x √ x n – a 2 = 2 an arccos a x n/2 . 4. dx x √ ax 2n + bx n =– 2 √ ax 2n + bx n bnx n . The parameters a, b, p, m, and n below in Integrals 5–8 can assume arbitrary values, except for those at which denominators vanish in successive applications of a formula. Notation: w = ax n + b. 5. x m (ax n + b) p dx = 1 m + np + 1 x m+1 w p + npb x m w p–1 dx . 6. x m (ax n + b) p dx = 1 bn(p + 1) –x m+1 w p+1 +(m + n + np + 1) x m w p+1 dx . 7. x m (ax n + b) p dx = 1 b(m + 1) x m+1 w p+1 – a(m + n + np + 1) x m+n w p dx . 8. x m (ax n + b) p dx = 1 a(m + np + 1) x m–n+1 w p+1 –b(m – n + 1) x m–n w p dx . . The parameters a, b, p, m, and n below in Integrals 5–8 can assume arbitrary values, except for those at which denominators vanish in successive applications of a formula. Notation: w = ax n +. x 2 ) –3/2 dx = a –2 x(a 2 – x 2 ) –1/2 . T2.1.2-6. Integrals involving arbitrary powers. Reduction formulas. 1. dx x(ax n + b) = 1 bn ln x n ax n + b . 2. dx x √ x n + a 2 = 2 an ln x n/2 √ x n +. 1130 INTEGRALS 2. dx (a + x)(b + x) = 1 a – b ln b + x a + x , a ≠ b.Fora = b, see Integral 2 with n =–2 in Paragraph T2.1.1-1. 3. xdx (a + x)(b + x) = 1 a – b a