First slide

Evaluation of definite integrals

Question

$\int_{0}^{1} x \tan^{- 1} x d x$

Moderate

A

$\frac{π}{4} - \frac{1}{2}$
B

$\frac{π}{18} - \frac{1}{2}$
C

$\frac{π}{4} + \frac{1}{2}$
D

$\frac{π}{8} + \frac{1}{2}$

Solution

$\begin{array}{l} \int_{0}^{1} x \tan^{- 1} x d x = {[\frac{x^{2}}{2} \cdot \tan^{- 1} x]}_{0}^{1} - \int_{0}^{1} \frac{1}{1 + x^{2}} \cdot \frac{x^{2}}{2} \cdot d x \\ = \frac{π}{8} - \frac{1}{2} \int_{0}^{1} (1 - \frac{1}{1 + x^{2}}) d x \\ = \frac{π}{8} - \frac{1}{2} {[x - \tan^{- 1} x]}_{0}^{1} \\ = \frac{π}{8} - \frac{1}{2} (1 - \frac{π}{4} - 0) \\ = \frac{π}{4} - \frac{1}{2} \end{array}$

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