1550251515-Classical_Complex_Analysis__Gonzalez_

Singularities/Residues/Applications 731

1

(^00) xa ln x 7rba

6. --b-dx = -.- 2 -(7rcosa7r -lnbsina7r), -1 <a< 0, b > O

o x + sm a7r

1.100 (lnx)2 dx = 2_71'3

0 x^2 +x+l 81v'3

8. (PV) loo ~x:~ = i7r2
   3.
   9
   1

00

(lnx) dx _ 7rlna [ 4 (ln ) 2 2 ]

· 2 2 -

8

a + 571' , a > 0

0 x +a a

1

(^00) lnxdx

10. o (x + 1)2JX = -71'


11. loo xdx = !71'2
    0 sinhx^4

1

(^00) sinax 1 1

12. ~h dx = -7rtanh -a'lf', a real

0 smx^2 2

1

(^00) cosh 2x 1 1

13. h dx=-7rsec-a7r,-l<a<l
    0 cos x 2 2

1

(^00) xcosax 1

14.. h dx = -^2 2 1
4

71' sech -a'lf', a real

0 sm x^2

1

(^00) cos ax dx 1 1

15. h = -
    2
    7rsech -
    2

a7r, -1<a<1

0 cos x

-1 { s

2
16. .C (s } 1.^1
2 + 4 ) 2 = 4 sm2t+ 2 tcos2t

{

1 } 4

00
(-1r-^1 1
17 .. .c-^1 h = 1-- '°" cos -(2n - l)7rt
s cos s 7r L.J 2n - 1 2
n=l
_^1 { 1 } 1 2 2 L

00

18. C (-1r
    ssms^2. h = - 2 t + 7r^2 --n^2 -(1-cosn7rt)
    n=l
    -1 { 1 } · 8

00

(-l)n. 1

19. C scoss 2 h = t + 71' 2 L (,n- 2 l) 2 sin - 2 (2n - 1 )7rt
    n=l

20. .c-l { coshas } =! (t^2 + a^2 b^2 ) 16b

2

~ (-l)n

s3 coshbs 2 7r^3 ~ (2n -1)^3

(2n-l)7ra (2n-l)7rt (O b)

cos

2

b cos

2

b < a <

21. = -71'
    l

oo dx 1

1 x../x^2 -1 2

22. f 1 dx ~ = 5:,

-1 (1+x^2 )^2 v1-x^2 8v2

f

23.^1 -r;:==d;::x::;:==::;= -
-1 {/(1 + x)^2 (1-x)