find-f-x-0-pi-4-ln-cost-xsint-dt-

Question Number 49938 by turbo msup by abdo last updated on 12/Dec/18

f i n d f (x) = \int_{0}^{\frac{π}{4}} l n (c o s t + x s i n t) d t

Commented by Abdo msup. last updated on 24/Dec/18

w e h a v e f^{^{'}} (x) = \int_{0}^{\frac{π}{4}} \frac{s i n t}{c o s t + x s i n t} d t

=_{t a n (\frac{t}{2}) = u} \int_{0}^{\sqrt{2} - 1} \frac{\frac{2 u}{1 + u^{2}}}{\frac{1 - u^{2}}{1 + u^{2}} + \frac{2 x u}{1 + u^{2}}} \frac{2 d u}{1 + u^{2}}

= \int_{0}^{\sqrt{2} - 1} \frac{4 u d u}{(1 + u^{2}) (1 - u^{2} + 2 x u)}

= - 4 \int_{0}^{\sqrt{2} - 1} \frac{u}{(u^{2} + 1) (u^{2} - 2 x u - 1)} d u l e t d e c o m p o s e

F (u) = \frac{u}{(u^{2} + 1) (u^{2} - 2 x u - 1)}

l e t u^{2} - 2 x u - 1

Δ^{^{'}} = x^{2} + 1 > 0 \Rightarrow u_{1} = x + \sqrt{1 + x^{2}}

u_{2} = x - \sqrt{1 + x^{2}}

F (u) = \frac{a u + b}{u^{2} + 1} + \frac{c}{u - u_{1}} + \frac{d}{u - u_{2}}

c = {l i m}_{u \to u_{1}} (u - u_{1}) F (u) = \frac{u_{1}}{(u_{1}^{2} + 1) (u_{1} - u_{2})}

= \frac{x + \sqrt{1 + x^{2}}}{({(x + \sqrt{1 + x^{2}})}^{2} + 1) 2 \sqrt{1 + x^{2}}}

d = {l i m}_{u \to u_{2}} (u - u_{2}) F (u) = \frac{u_{2}}{(u_{2}^{2} + 1) (- 2 \sqrt{1 + x^{2}})}

= - \frac{u_{2}}{({(x - \sqrt{1 + x^{2}})}^{2} + 1) 2 \sqrt{1 + x^{2}}} \dots b e c o n t i n u e d \dots