Q73 / 150 · MHT CET 2016 · Mathematics

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

A
$\log a \cdot a^{x + \tan^{- 1} x} + c$
B
$\frac{(x + \tan^{- 1} x)}{\log a} + c$
Correct answer
C
$\frac{a^{x + \tan^{- 1} x}}{\log a} + c$
D
$\log a \cdot (x + \tan^{- 1} x) + c$

Solution

I = \int \frac{(x^{2} + 2) a^{(x + \tan^{- 1} x)}}{x^{2} + 1} d x = \int (1 + \frac{1}{x^{2} + 1}) e^{(x + \tan^{- 1} x) \cdot l n a} d x

Let

(x + \tan^{- 1} x) l n a = t

\Rightarrow (1 + \frac{1}{1 + x^{2}}) d x = \frac{d t}{l n a}

\Rightarrow I = \int \frac{e^{t} l t}{l n a} = \frac{e^{t}}{l n a} + C

= \frac{a^{x + \tan^{- 1} x}}{l n a} + C

Hence, the correct answer is option: 'C'.

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