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Question Number 53462 by maxmathsup by imad last updated on 22/Jan/19

f i n d \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{x s i n x}{{c o s}^{2} x} d x

Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19

\int x t a n x s e c x d x

x \int t a n x s e c x d x - \int [\frac{d x}{d x} \int t a n x s e c x d x] d x

x s e c x - \int s e c x d x

x s e c x - l n (s e c x + t a n x) + c

s o I =∣ x s e c x - l n (s e c x + t a n x) ∣_{\frac{- π}{4}}^{\frac{π}{4}}

= [{\frac{π}{4} \times \sqrt{2} - l n (\sqrt{2} + 1)} - {\frac{- π}{4} \times \sqrt{2} - l n (\sqrt{2} - 1)}]

= [2 \times \frac{π}{4} \times \sqrt{2} + l n (\frac{\sqrt{2} - 1}{\sqrt{2} + 1})]

= \frac{π}{\sqrt{2}} + l n (\frac{\sqrt{2} - 1}{\sqrt{2} + 1})