find-the-value-of-0-pi-2-ln-cos-d-and-0-pi-2-ln-sin-d-

Question Number 26357 by abdo imad last updated on 24/Dec/17

f i n d t h e v a l u e o f \int_{0}^{\frac{π}{2}} l n (c o s θ) d θ a n d

\int_{0}^{\frac{π}{2}} l n (s i n θ) d θ .

Commented by prakash jain last updated on 24/Dec/17

\int_{0}^{π / 2} \ln (\cos x) d x

u = \frac{π}{2} - x

d u = - d x

I = \int_{0}^{π / 2} \ln (\sin x) d x = - \int_{π / 2}^{0} \ln (\cos u) d u

= \int_{0}^{π / 2} \ln (\cos x) d x

2 I = \int_{0}^{π / 2} \ln (\cos x \sin x) d x

= \int_{0}^{π / 2} \ln \sin 2 x d x - \int_{0}^{π / 2} \ln 2 d x

= I - \frac{π}{2} \ln 2

I = - \frac{π}{2} \ln 2

Answered by prakash jain last updated on 24/Dec/17

- \frac{π}{2} \ln 2