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Question Number 55267 by maxmathsup by imad last updated on 25/Feb/19

1) c a l c u l a t e f (x) = \int_{0}^{\frac{π}{4}} l n (1 + x t a n θ) d θ

2) f i n d t h e v a l u e s o f i n t e g r a l s \int_{0}^{\frac{π}{4}} l n (1 + t a n θ) a n d \int_{0}^{\frac{π}{4}} l n (1 + 2 t a n θ) d θ .

1) w e h a v e f^{^{'}} (x) = \int_{0}^{\frac{π}{4}} \frac{t a n θ}{1 + x t a n θ} d θ = \int_{0}^{\frac{π}{4}} \frac{\frac{s i n θ}{c o s θ}}{1 + x \frac{s i n θ}{c o s θ}} d θ

= \int_{0}^{\frac{π}{4}} \frac{s i n θ}{c o s θ + x s i n θ} d θ =_{t a n (\frac{θ}{2}) = t} \int_{0}^{\sqrt{2} - 1} \frac{\frac{2 t}{1 + t^{2}}}{\frac{1 - t^{2}}{1 + t^{2}} + \frac{2 x t}{1 + t^{2}}} \frac{2 d t}{1 + t^{2}}

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

F (t) = \frac{4 t}{(t^{2} + 1) (t^{2} - 2 x t - 1)} r o o t s o f t^{2} - 2 x t - 1

Δ^{^{'}} = x^{2} + 1 \Rightarrow t_{1} = x + \sqrt{x^{2} + 1} a n d t_{2} = x - \sqrt{x^{2} + 1}

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

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

b = {l i m}_{t \to t_{2}} (t - t_{2}) F (t) = \frac{4 t_{2}}{(t_{2}^{2} + 1) (t_{2} - t_{1})} = β \Rightarrow F (t) = \frac{α}{t - t_{1}} + \frac{β}{t - t_{2}} + \frac{c t + d}{t^{2} + 1}

F (0) = 0 = - \frac{α}{t_{1}} - \frac{β}{t_{2}} + d \Rightarrow d = \frac{α}{t_{1}} + \frac{β}{t_{2}}

F (1) = \frac{2}{- 2 x} = - \frac{1}{x} = \frac{α}{1 - t_{1}} + \frac{β}{1 - t_{2}} + \frac{c + d}{2} \Rightarrow \frac{1}{x} = \frac{α}{t_{1} - 1} + \frac{β}{t_{2} - 1} - \frac{c}{2} - \frac{d}{2}

\Rightarrow \frac{c}{2} = \frac{α}{t_{1} - 1} + \frac{β}{t_{2} - 1} - \frac{d}{2} - \frac{1}{x} \Rightarrow c = \frac{2 α}{t_{1} - 1} + \frac{2 β}{t_{2} - 1} - d - \frac{2}{x}

\int F (t) d t = α l n ∣ t - t_{1} ∣ + β l n ∣ t - t_{2} ∣ + \frac{c}{2} l n (t^{2} + 1) + d a r c t a n (t) \Rightarrow

\int_{0}^{\sqrt{2} - 1} F (t) d t = {[α l n ∣ t - t_{1} ∣ + β l n ∣ t - t_{2} ∣ + \frac{c}{2} l n (t^{2} + 1)]}_{0}^{\sqrt{2} - 1}

= α l n ∣ \sqrt{2} - 1 - t_{1} ∣ + β l n ∣ \sqrt{2} - 1 - t_{2} ∣ + \frac{c}{2} l n (4 - 2 \sqrt{2})

= α l n ∣ \sqrt{2} - 1 - x - \sqrt{1 + x^{2}}) + β l n ∣ \sqrt{2} - 1 - x + \sqrt{1 + x^{2}}) + \frac{l n (4 - 2 \sqrt{2})}{2} c = f^{^{'}} (x) \Rightarrow

f (x) = \int α l n ∣ \sqrt{2} - 1 - x - \sqrt{1 + x^{2}} ∣) d x + β \int l n ∣ \sqrt{2} - 1 + \sqrt{1 + x^{2}} ∣ d x

+ \frac{c x}{2} l n (4 - 2 \sqrt{2}) + C \dots . b e c o n t i n u e d \dots