First slide

Methods of integration

Question

$\int e^{x} (\frac{1 + \sin x}{1 + \cos x}) dx$ is equal to

Moderate

A

$e^{x} \sec \frac{x}{2} + C$
B

$\frac{e^{x}}{1 + \cos x} + C$
C

$e^{x} \tan \frac{x}{2} + C$
D

None of these

Solution

$\int e^{x} (\frac{1}{1 + \cos x} + \frac{\sin x}{1 + \cos x}) dx = \int e^{x} (\frac{1}{2 \cos^{2} \frac{x}{2}} + \frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos^{2} \frac{x}{2}}) dx (∵ \cos 2 x = 2 \cos^{2} x - 1 and \sin 2 x = 2 \sin xcos x) = \int e^{x} (\frac{\sec^{2} \frac{x}{2}}{2} + \tan \frac{x}{2}) dx = \int e^{x} (\tan \frac{x}{2} + \frac{1}{2} \sec^{2} \frac{x}{2}) dx Let f (x) = \tan \frac{x}{2} \Rightarrow f^{'} (x) = \frac{\sec^{2} \frac{x}{2}}{2} ∴ \int e^{x} (\tan \frac{x}{2} + \frac{\sec^{2} \frac{x}{2}}{2}) dx = e^{x} \tan \frac{x}{2} + C [∵ \int e^{x} [f (x) + f^{'} (x)] dx = e^{x} f (x)]$

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