calculate-0-pi-4-cos-x-ln-cos-x-dx-

Question Number 32353 by abdo imad last updated on 23/Mar/18

c a l c u l a t e \int_{0}^{\frac{π}{4}} c o s (x) l n (c o s (x)) d x .

Answered by sma3l2996 last updated on 25/Mar/18

A = \int_{0}^{π / 4} c o s (x) l n (c o s x) d x

b y p a r t s

u = l n (c o s x) \Rightarrow u^{'} = - t a n x

v^{'} = c o s x \Rightarrow v = s i n x

A = {[s i n (x) l n (c o s x)]}_{0}^{π / 4} + \int_{0}^{π / 4} \frac{{s i n}^{2} x}{c o s x} d x = - \frac{\sqrt{2}}{4} l n (2) + \int_{0}^{π / 4} (\frac{1}{c o s x} - c o s x) d x

l e t t = t a n (x / 2)

A = - \frac{\sqrt{2}}{4} l n (2) - {[s i n x]}_{0}^{π / 4} + \int_{0}^{t a n (π / 8)} \frac{d t}{1 - t^{2}}

= - \frac{\sqrt{2}}{4} l n (2) - \frac{\sqrt{2}}{2} + \frac{1}{2} {[l n ∣ \frac{1 + t}{1 - t} ∣]}_{0}^{t a n (π / 8)}

A = - \frac{\sqrt{2}}{4} l n (2) - \frac{\sqrt{2}}{2} + \frac{1}{2} l n (\frac{2}{1 - t a n (π / 8)} - 1)