$\int (\frac{(x^{2} + 2) a^{(x + {tan}^{- 1} x)}}{x^{2} + 1}) d x =$ ........

log a . a^{(x + {tan}^{- 1} x)} + c

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\frac{x + {tan}^{- 1} x}{log a} + c

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\frac{a^{(x + {tan}^{- 1} x)}}{log a} + c

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log a . (x + {tan}^{- 1} x) + c

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Solution

The correct option is D $\frac{a^{(x + {tan}^{- 1} x)}}{log a} + c$
$\int \frac{(x^{2} + 2) a^{(x + {t a n}^{- 1} x)}}{x^{2} + 1} d x$
Let $x + {t a n}^{- 1} x = t \Rightarrow (1 + \frac{1}{x^{2} + 1}) d x = d t \Rightarrow (\frac{x^{2} + 2}{x^{2} + 1}) d x = d t ∴ \int \frac{(x^{2} + 2) a^{(x + {t a n}^{- 1} x)}}{x^{2} + 1} d x = \int a^{t} d t = \frac{a^{t}}{l o g a} + c = \frac{a^{(x + {t a n}^{- 1} x)}}{l o g a} + c$

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