calculate-0-pi-x-dx-1-cosx-

Question Number 36937 by maxmathsup by imad last updated on 07/Jun/18

c a l c u l a t e \int_{0}^{π} \frac{x d x}{1 + c o s x}

Commented by math khazana by abdo last updated on 08/Jun/18

I = \int_{0}^{\frac{π}{2}} \frac{x d x}{1 + c o s x} + \int_{\frac{π}{2}}^{π} \frac{x d x}{1 + c o s x} b u t

\int_{\frac{π}{2}}^{π} \frac{x d x}{1 + c o s x} =_{x = \frac{π}{2} + t} \int_{0}^{\frac{π}{2}} \frac{(\frac{π}{2} + t) d t}{1 - s i n t}

= \frac{π}{2} \int_{0}^{\frac{π}{2}} \frac{d t}{1 - s i n t} + \int_{0}^{\frac{π}{2}} \frac{t s i n t}{1 - s i n (t)} d t

\int_{0}^{\frac{π}{2}} \frac{x d x}{1 + c o s x} =_{t a n (\frac{x}{2}) = t} \int_{0}^{1} \frac{2 a r c t a n (t)}{1 + \frac{1 - t^{2}}{1 + t^{2}}} \frac{2 d t}{1 + t^{2}}

= 4 \int_{0}^{1} \frac{a r c t a n (t)}{1 + t^{2} + 1 - t^{2}} d t = 2 \int_{0}^{1} a r c t a n t d t

= 2 {{[t a r c t a n (t)]}_{0}^{1} - \int_{0}^{1} \frac{t}{1 + t^{2}} d t}

= 2 {\frac{π}{4} - \frac{1}{2} {[l n (1 + t^{2})]}_{0}^{1}} = \frac{π}{2} - l n (2)

\int_{0}^{\frac{π}{2}} \frac{t s i n (t)}{1 - s i n (t)} d t =_{t a n (\frac{t}{2}) = x} \int_{0}^{1} \frac{2 a r c t a n t \frac{2 t}{1 + t^{2}}}{1 - \frac{2 t}{1 + t^{2}}} d t

= 4 \int_{0}^{1} \frac{t a r c t a n (t)}{1 + t^{2} - 2 t} d t = 4 \int_{0}^{1} \frac{t a r c t a n (t)}{{(t - 1)}^{2}} d t

b y p a r t s \int_{0}^{1} \frac{t a r c t a n (t)}{{(t - 1)}^{2}} d t

= \int_{0}^{1} \frac{(t - 1 + 1) a r c t a n (t)}{{(t - 1)}^{2}} d t

= \int_{0}^{1} \frac{a r c t a n t}{t - 1} d t + \int_{0}^{1} \frac{a r c t a n (t)}{{(t - 1)}^{2}} d t b u t

\int_{0}^{1} \frac{a r c t a n (t)}{1 - t} d t = \int_{0}^{1} (\sum_{n = 0}^{\infty} t^{n}) a r c t a n t d t

= \sum_{n = 0}^{\infty} \int_{0}^{1} t^{n} a r c t a n (t) d t = \sum_{n = 0}^{\infty} A_{n}

A_{n} = t^{n} a r c t a n t \sim A t^{n} b e c a u s e a r c t a n i s b o r n e d

\Rightarrow \int_{0}^{1} t^{n} a r c t a n t d t \sim \frac{A}{n + 1} a n d Σ A_{n} d i v e r g e s

s o t h i s i n t e g r a l d i v e r g e s a n d w e d o t h e s a m e

m a n n e r f o r t h e o t h e r i n t e g r a l s o I d i v e r g e s . .!