prove that integrate 0 to a √x / √x + √a-x = pie/4
Answers
Z ∞
0
sin x
x
dx =
π
2
Proof. We note by the symmetry of f(x) = 1
x
sin x, 2 R ∞
0
sin x
x =
R +∞
−∞
sin x
x
. We also know that R R
−R
sin x
x
dx =
Im R R
−R
e
ix
ix , and so we only have to calculate
lim
→0,R→∞ Z
l−
e
ix
x
dx +
Z
l+
e
ix
x
dx (1)
where we are using the contours shown below. By Cauchy’s theorem, we can conclude that
Z ∞
−∞
e
ix
x
dx =
Z
c
e
iz
z
dz +
Z
cR
e
z Z ∞
0
sin x
x
dx =
π
2
Proof. We note by the symmetry of f(x) = 1
x
sin x, 2 R ∞
0
sin x
x =
R +∞
−∞
sin x
x
. We also know that R R
−R
sin x
x
dx =
Im R R
−R
e
ix
ix , and so we only have to calculate
lim
→0,R→∞ Z
l−
e
ix
x
dx +
Z
l+
e
ix
x
dx (1)
where we are using the contours shown below. By Cauchy’s theorem, we can conclude that
Z ∞
−∞
e
ix
x
dx =
Z
c
e
iz
z
dz +
Z
cR
e
z
z
dz
z
dz