calculate-0-pi-3-sinx-cos-cosx-1-2sin-cosx-dx-

Question Number 35588 by abdo mathsup 649 cc last updated on 20/May/18

c a l c u l a t e \int_{0}^{\frac{π}{3}} \frac{s i n x c o s (c o s x)}{1 + 2 s i n (c o s x)} d x

Commented by abdo imad last updated on 21/May/18

w e r e m a r k t h a t {(s i n (c o s x))}^{^{'}} = - s i n x c o s (c o s x) \Rightarrow

I = - \frac{1}{2} \int_{0}^{\frac{π}{3}} \frac{{(1 + 2 s i n (c o s x))}^{^{'}}}{1 + 2 s i n (c o s x)} d x

= - \frac{1}{2} {[l n ∣ 1 + 2 s i n (c o s x) ∣]}_{0}^{\frac{π}{3}} =

= - \frac{1}{2} {l n (1 + 2 s i n (\frac{1}{2}) - l n (1 + 2 s i n (1)}

Answered by RAMANUJAN last updated on 20/May/18

[\ln {(2 Π + 3) / 3}] / 2 - 0

is the correct answer