BeMath-1-find-1-2-2-0-pi-2-x-sin-x-1-cos-x-2-dx-

Question Number 108450 by bemath last updated on 17/Aug/20

\frac{B e M a t h}{\subset\supset}

(1) f i n d (\frac{1}{2})!

(2) \underset{0}{\int^{π / 2}} \frac{x \sin x}{{(1 + \cos x)}^{2}} d x

Commented by Smail last updated on 17/Aug/20

I_{n} = \int_{0}^{\infty} t^{n} e^{- t} d t = n!

n = \frac{1}{2}

I_{1 / 2} = \int_{0}^{\infty} \sqrt{t} e^{- t} d t = (\frac{1}{2})!

x = \sqrt{t} \Rightarrow d x = \frac{d t}{2 \sqrt{t}}

I_{1 / 2} = 2 \int_{0}^{\infty} x^{2} e^{- x^{2}} d x = (\frac{1}{2})!

B y p a r t s

u = x \Rightarrow u^{'} = 1

v^{'} = {x e}^{- x^{2}} \Rightarrow v = \frac{- 1}{2} e^{- x^{2}}

I_{1 / 2} = \int_{0}^{\infty} e^{- x^{2}} d x = (\frac{1}{2})!

\int_{0}^{\infty} e^{- x^{2}} d x = \frac{\sqrt{π}}{2}

S o

(\frac{1}{2})! = \frac{\sqrt{π}}{2}

Commented by bemath last updated on 17/Aug/20

t h a n k y o u

Answered by john santu last updated on 17/Aug/20

Answered by Dwaipayan Shikari last updated on 17/Aug/20

(\frac{1}{2})! = \frac{1}{2} Γ (\frac{1}{2}) = \frac{\sqrt{π}}{2}