First slide

Methods of integration

Question

$\int \frac{2 x + \sqrt{\tan^{- 1} x}}{1 + x^{2}} d x =$

Moderate

A

$\log (1 + x^{2})$
B

$\log (1 + x^{2}) + {(\tan^{- 1} x)}^{\frac{3}{2}} + C$
C

$\log (1 + x^{2}) + {(\cot^{- 1} x)}^{\frac{3}{2}} + C$
D

$\log (1 + x^{2}) + \frac{2}{3} {(t a n^{- 1} x)}^{\frac{3}{2}} + C$

Solution

$\begin{array}{l} I = \int \frac{2 x + \sqrt{\tan^{- 1} x}}{1 + x^{2}} d x \\ I = \int \frac{2 x}{1 + x^{2}} d x + \int \frac{\sqrt{\tan^{- 1} x}}{1 + x^{2}} d x \\ = \int \frac{2 x}{1 + x^{2}} d x + \int \sqrt{t} d t ∵ \tan^{- 1} x = t, \frac{1}{1 + x^{2}} d x = d t \\ = \log (1 + x^{2}) + \frac{2}{3} t^{\frac{3}{2}} + C \\ = \log (1 + x^{2}) + \frac{2}{3} {(\tan^{- 1} x)}^{\frac{3}{2}} + C \end{array}$

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