Q113 / 160 · AP EAMCET 2013 · Mathematics

$\int e^x\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right) d x$ is equal to

  • A
    $e^x \cot x+C$
  • B
    $2 e^x \sec ^2 x+C$
  • C
    $e^x \cos 2 x+C$
  • Correct answer
    D
    $e^x \tan x+C$

Solution

Let $\begin{aligned} I & =\int e^x\left(\frac{2+\sin 2 x}{1+\cos 2 x}\right) d x \\ & =\int e^x\left\{\frac{2}{1+\cos 2 x}+\frac{\sin 2 x}{1+\cos 2 x}\right\} d x \\ & =\int e^x\left\{\frac{2}{2 \cos ^2 x}+\frac{2 \sin x \cos x}{2 \cos ^2 x}\right\} d x \\ & =\int e^x\left(\sec ^2 x+\tan x\right) d x \\ & =\int e^x \sec ^2 x d x+\int e^x \tan x d x \\ & =e^x \tan x-\int e^x \tan x d x+\int e^x \tan x d x \\ & =e^x \tan x+C\end{aligned}$

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

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