$\int e^{x / 2} \sec 2 x (1 + 4 \tan 2 x) dx$ is equal to

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$\frac{1}{2} e^{x / 2} Sec 2 x + C$

$4 e^{x / 2} Sec 2 x + C$

$e^{x / 2} Sec 2 x + C$

$2 e^{x / 2} Sec 2 x + C$

answer is B.

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Detailed Solution

$\begin{array}{l} \int e^{x / 2} \sec 2 xdx = \sec 2 x \frac{e^{x / 2}}{(1 / 2)} - \int 2 \sec 2 xtan 2 x \frac{e^{x / 2}}{1 / 2} dx \\ = \sec 2 {xe}^{x / 2} - \int 4 e^{x / 2} \sec 2 xtan 2 xdx \end{array}$

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