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Question Number 40044 by abdo mathsup 649 cc last updated on 15/Jul/18

l e t f (t) = \int_{0}^{\frac{π}{2}} l n (c o s x + t s i n x)

1) c a l c u l a t e f (0)

2) c a l c u l a t e f^{^{'}} (t) t h e n f i n d a s i m p l e f o r m o f f (t)

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

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

Commented by abdo mathsup 649 cc last updated on 15/Jul/18

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

Commented by maxmathsup by imad last updated on 22/Jul/18

1) w e h a v e f (t) = \int_{0}^{\frac{π}{2}} l n (c o s x + t s i n x) d x \Rightarrow f (0) = \int_{0}^{\frac{π}{2}} l n (c o s x) d x = - \frac{π}{2} l n (2)

(t h i s v a l u e i s p r o v e d)

2) f^{^{'}} (t) = \int_{0}^{\frac{π}{2}} \frac{\partial}{\partial t} {l n (c o s x + t s i n x)} d x

= \int_{0}^{\frac{π}{2}} \frac{s i n x}{c o s x + t s i n x} d x h a n g e m e n t t a n (\frac{x}{2}) = u g i v e

f^{^{'}} (t) = \int_{0}^{1} \frac{\frac{2 u}{1 + u^{2}}}{\frac{1 - u^{2}}{1 + u^{2}} + t \frac{2 u}{1 + u^{2}}} \frac{2 d u}{1 + u^{2}} = \int_{0}^{1} \frac{4 u d u}{(1 + u^{2}) {1 - u^{2} + 2 t u}}

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

r o o t s o f u^{2} - 2 t u - 1

Δ^{^{'}} = t^{2} + 1 \Rightarrow u_{1} = t + \sqrt{1 + t^{2}} a n d u_{2} = t - \sqrt{1 + t^{2}} a n d F (u) = \frac{4 u}{(u - u_{1}) (u - u_{2}) (1 + u^{2})}

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

a = {l i m}_{u \to u_{1}} (u - u_{1}) F (u) = \frac{4 u_{1}}{(u_{1} - u_{2}) (1 + u_{1}^{2})} = \frac{4 (t + \sqrt{1 + t^{2}})}{2 \sqrt{1 + t^{2}} (1 + {(t + \sqrt{1 + t^{2}})}^{2}}

b = {l i m}_{u \to u_{2}} (u - u_{2}) F (u) = \frac{4 u_{2}}{(u_{2} - u_{1}) (1 + u_{2}^{2})} = \frac{4 (t - \sqrt{1 + t^{2})}}{- 2 \sqrt{1 + t^{2}} {1 + {(t - \sqrt{1 + t^{2}})}^{2}}

{l i m}_{u \to + \infty} u F (u) = 0 = a + b + c \Rightarrow c = - a - b \Rightarrow

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

F (0) = 0 = - \frac{a}{u_{1}} - \frac{b}{u_{2}} + d \Rightarrow d = \frac{a}{u_{1}} + \frac{b}{u_{2}} \dots . b e c o n t i n u e d \dots