First slide

Methods of integration

Question

$\int \frac{d x}{5 \sin x + 12 \cos x}$

Moderate

A

$\frac{1}{13} \log |\tan (\frac{x}{2} + \frac{1}{2} \tan^{- 1} (\frac{12}{5}))| + C$
B

$\log |\tan (\frac{x}{2} + \frac{1}{2} \tan^{- 1} (\frac{12}{5}))| + C$
C

$\frac{1}{13} \log |\tan (\frac{x}{2} - \frac{1}{2} \tan^{- 1} (\frac{12}{5}))| + C$
D

None

Solution

$\frac{1}{r} \int \frac{d x}{\sin (x + α)}$ $\begin{matrix} r \cos α = 5 \\ r \sin α = 12 \end{matrix}$
$= \frac{1}{r} \log |\tan (\frac{x + α}{2})| + C r = 13 \tan α = \frac{12}{5}$

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