Page 724 - Handbook Of Integral Equations
P. 724
1 n
2ln2 4(–1) n (–1) k
9. x n–1/2 ln(1 + x) dx = + π – , n =1, 2, ...
2n +1 2n +1 2k +1
0 k=0
∞ a + x
2 2
10. ln 2 2 dx = π(a – b), a, b >0.
0 b + x
p–1 2
∞ x ln x π cos(πp/q)
11. q dx = – 2 2 , 0 < p < q.
0 1+ x q sin (πp/q)
∞ 1
12. e –µx ln xdx = – (C +ln µ), µ >0, C = 0.5772 ...
µ
0
3.5. Integrals Containing Trigonometric Functions
π/2
π 1 ⋅ 3 ... (2n – 1)
1. cos 2n xdx = , n =1, 2, ...
2 2 ⋅ 4 ... (2n)
0
π/2 2 ⋅ 4 ... (2n)
2. cos 2n+1 xdx = , n =1, 2, ...
0 1 ⋅ 3 ... (2n +1)
π/2 m–1
(n – 2k + 1)(n – 2k +3) ... (n – 1) 1
n
3. x cos xdx = –
0 (n – 2k)(n – 2k +2) ...n n – 2k
k=0
π (2m – 2)!!
if n =2m – 1,
2 (2m – 1)!!
+ 2 m =1, 2, ...
π (2m – 1)!!
⋅ if n =2m,
8 (2m)!!
n
π dx π (2n – 2k – 1)!! (2k – 1)!! a + b k
4. n+1 = √ , a > |b|.
2
n
0 (a + b cos x) 2 (a + b) n a – b 2 (n – k)! k! a – b
k=0
∞ cos ax π
5. √ dx = , a >0.
0 x 2a
∞
cos ax – cos bx
6. dx =ln , ab ≠ 0.
b
0 x a
∞ cos ax – cos bx
1
7. 2 dx = π(b – a), a, b ≥ 0.
2
0 x
∞
–µ
8. x µ–1 cos ax dx = a Γ(µ) cos 1 πµ , a >0, 0 < µ <1.
2
0
∞
cos ax π –ab
9. dx = e , a, b >0.
2
b + x 2 2b
0
√
∞
cos ax π 2 ab ab ab
10. 4 4 dx = 3 exp – √ cos √ + sin √ , a, b >0.
0 b + x 4b 2 2 2
∞ cos ax π
11. 2 2 2 dx = 3 (1 + ab)e –ab , a, b >0.
0 (b + x ) 4b
∞ cos ax dx π be – ce
–ac –ab
12. 2 2 2 2 = , a, b, c >0.
2
0 (b + x )(c + x ) 2bc b – c 2
∞
2 1 π
13. cos ax dx = , a >0.
0 2 2a
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
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