advanced-calculus-calculate-0-pi-pi-1-sin-x-cos-x-dx-

Question Number 123020 by mnjuly1970 last updated on 21/Nov/20

\dots a d v a n c e d c a l c u l u s . .

c a l c u l a t e :: \emptyset = \int_{0}^{π} \frac{π}{1 - s i n (x) c o s (x)} d x = ? ? ?

\dots \dots \dots \dots \dots \dots . .

Answered by Dwaipayan Shikari last updated on 21/Nov/20

\int_{0}^{π} \frac{π}{1 - \frac{1}{2} s i n 2 x} d x

= 2 \int_{0}^{\infty} \frac{2 π}{2 - \frac{2 t}{1 + t^{2}}} . \frac{1}{1 + t^{2}} d t t = t a n x

= 2 \int_{0}^{\infty} \frac{π}{t^{2} - t + 1} d t = 2 \int_{0}^{\infty} \frac{π}{(t - \frac{1}{2}) + {(\sqrt{\frac{3}{4}})}^{2}} = {[\frac{4 π}{\sqrt{3}} {t a n}^{- 1} \frac{2 t - 1}{\sqrt{3}}]}_{0}^{\infty} = \frac{2 π^{2}}{\sqrt{3}}

Commented by mnjuly1970 last updated on 21/Nov/20

t h a n k y o u s o m u c h \dots