$\int \frac{{(1 + x)}^{2}}{x + x^{3}} d x$ is equal to

{log}_{e} x + {log}_{e} (1 + x^{2}) + k

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{log}_{e} x + {tan}^{- 1} x + k

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{log}_{e} x + 2 {tan}^{- 1} x + k

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none of these

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Solution

The correct option is C ${log}_{e} x + 2 {tan}^{- 1} x + k$
$\int \frac{{(1 + x)}^{2}}{x + x^{3}} d x$
$= \int \frac{x^{2} + 2 x + 1}{x (1 + x^{2})} d x$
$\frac{x^{2} + 2 x + 1}{x (1 + x^{2})} = \frac{1}{x} + \frac{2}{(1 + x^{2})}$
$\int \frac{x^{2} + 2 x + 1}{x (1 + x^{2})} d x = {log}_{e} x + 2 {tan}^{- 1} x + k$

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