First slide

Evaluation of definite integrals

Question

$\int_{0}^{1} T a n^{- 1} (\frac{2 x}{1 - x^{2}}) d x =$

Easy

A

$\frac{π}{2} - l n 2$
B

$π - l n 2$
C

$π + l n 2$
D

$\frac{π}{4}$

Solution

$\int_{0}^{1} T a n^{- 1} (\frac{2 x}{1 - x^{2}}) d x = \int_{0}^{1} 2 T a n^{- 1} x d x$
              $= 2 [{(T a n^{- 1} x . x)}_{0}^{1}] - 2 \int_{0}^{1} \frac{1}{1 + x^{2}} . x d x$

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

              $= \frac{π}{2} - {(\log (1 + x^{2}))}_{0}^{1}$

               $= \frac{π}{2} - \log 2$

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