Page 713 - Handbook Of Integral Equations

q
q
8. (ln x) dx = x(ln x) – q (ln x) q–1 dx, q ≠ –1.
m
x n+1 (–1) k
n
m
9. x (ln x) dx = (m +1)m... (m – k + 1)(ln x) m–k , n, m =1, 2, ...
m +1 (n +1) k+1
k=0
1 p+1 q q p q–1

p
q
10. x (ln x) dx = x (ln x) – x (ln x) dx, p, q ≠ –1.
p +1 p +1
1

11. ln(a + bx) dx = (ax + b) ln(ax + b) – x.
b
2 2
1 2 a 1 x a
12. x ln(a + bx) dx = x – ln(a + bx) – – x .
2 b 2 2 2 b
1 3 a 1 x ax a x
3 3 2 2
2
13. x ln(a + bx) dx = x – ln(a + bx) – – + .
3 b 3 3 3 2b b 2

ln xdx ln x 1 x
14. = – + ln .
(a + bx) 2 b(a + bx) ab a + bx

ln xdx ln x 1 1 x
15. = – + + ln .
2
(a + bx) 3 2b(a + bx) 2 2ab(a + bx) 2a b a + bx
√
 √
 2 √ √ a + bx + a
 (ln x – 2) a + bx + a ln √ √ if a >0,

ln xdx  b a + bx – a
16. √ = √
a + bx  2 √ √ a + bx

(ln x – 2) a + bx +2 –a arctan √ if a <0.


b –a

2
2
2
2
17. ln(x + a ) dx = x ln(x + a ) – 2x +2a arctan(x/a).

2
2
2
2
2
2
2

18. x ln(x + a ) dx = 1 (x + a ) ln(x + a ) – x .
2

2
2
2
3
3
2
2
3
2
2

19. x ln(x + a ) dx = 1 x ln(x + a ) – x +2a x – 2a arctan(x/a) .
3 3
2.6. Integrals Containing Trigonometric Functions
Integrals containing cos x. Notation: n =1, 2, ...
1

1. cos(a + bx) dx = sin(a + bx).
b

2. x cos xdx = cos x + x sin x.

2
2
3. x cos xdx =2x cos x +(x – 2) sin x.
n n–1
x 2n–2k x 2n–2k–1
4. x 2n cos xdx =(2n)! (–1) k sin x + (–1) k cos x .
(2n – 2k)! (2n – 2k – 1)!
k=0 k=0
n 2n–2k+1 2n–2k
k x x
2n+1
5. x cos xdx =(2n + 1)! (–1) sin x + cos x .
(2n – 2k + 1)! (2n – 2k)!
k=0

p
p
6. x cos xdx = x sin x + px p–1 cos x – p(p – 1) x p–2 cos xdx.

2
1
7. cos xdx = x + 1 sin 2x.
2 4
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 697