First slide

Methods of integration

Question

$\int \frac{(e^{x} + e^{- x}) + (e^{x} - e^{- x}) \sin x}{1 + \cos x} d x =$

Moderate

A

$(e^{x} + e^{- x}) \tan (x / 2) + C$
B

$(e^{x} - e^{- x}) \cot (x / 2) + C$
C

$(e^{x} - e^{- x}) \sec (x / 2) + C$
D

$(e^{x} - e^{- x}) cosec (x / 2) + C$

Solution

$\begin{array}{l} I = \int \frac{e^{x} + e^{- x} + (e^{x} - e^{- x}) \sin x}{1 + \cos x} dx \\ = \int (\frac{e^{x} + e^{- x}}{1 + \cos x} + \frac{(e^{x} - e^{- x}) \sin x}{1 + \cos x}) dx \\ = \int ((e^{x} + e^{- x}) \frac{1}{2} \sec^{2} \frac{x}{2} + (e^{x} - e^{- x}) \tan \frac{x}{2}) dx \\ = \int \frac{d}{dx} ((e^{x} + e^{- x}) \tan \frac{x}{2}) dx \\ = (e^{x} + e^{- x}) \tan (x / 2) + C \end{array}$

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