0-pi-2-sin-2xlog-tan-x-dx-

Question Number 152275 by peter frank last updated on 27/Aug/21

\int_{0}^{\frac{π}{2}} \sin 2 xlog (\tan x) dx

Answered by qaz last updated on 27/Aug/21

\int_{0}^{π / 2} \sin 2 x \cdot lntan xdx

= 2 \int_{0}^{π / 2} \sin xcos x (lnsin x - lncos x) dx

= {\int lnsin xd (\sin^{2} x) + \int lncos xd (\cos^{2} x)}_{0}^{π / 2}

= \sin^{2} xlnsin x + \cos^{2} xlncos x ∣_{0}^{π / 2}

= 0

Answered by mnjuly1970 last updated on 27/Aug/21

I = \int_{0}^{\frac{π}{2}} s i n (2 x) l n (t a n (x)) d x \dots (1)

I = \int_{0}^{\frac{π}{2}} s i n (2 x) . l n (c o t (x)) d x \dots (2)

(1) + (2) : 2 I = \int_{0}^{\frac{π}{2}} s i n (2 x) l n (1) = 0

I = 0 \dots