First slide

Methods of integration

Question

$Evaluate \int \frac{\tan x}{a + b \tan^{2} x} d x$

Moderate

A

$= \frac{1}{2 (b + a)} \log |a \cos^{2} x + b \sin^{2} x| + C$
B

$= \frac{1}{2 (b - a)} \log |a \cos^{2} - b \sin^{2} x| + C$
C

$= \frac{1}{(b - a)} \log |a \cos^{2} x + b \sin^{2} x| + C$
D

$= \frac{1}{2 (b - a)} \log |a \cos^{2} x + b \sin^{2} x| + C$

Solution

$\begin{array}{l} I = \int \frac{\tan x}{a + b \tan^{2} x} d x \\ = \int \frac{\frac{\sin x}{\cos x}}{a + b \frac{\sin^{2} x}{\cos^{2} x}} d x \\ = \int \frac{\sin x \cos x}{a \cos^{2} x + b \sin^{2} x} d x \\ = \frac{1}{2} \int \frac{\sin 2 x}{a \cos^{2} x + b \sin^{2} x} d x \\ \begin{array}{l} Put a \cos^{2} x + b \sin^{2} x = t \\ ∴ (b - a) \sin 2 x d x = d t \\ ∴ l = \frac{1}{2 (b - a)} \int \frac{d t}{t} \\ = \frac{1}{2 (b - a)} \log_{e} | t | + C \\ = \frac{1}{2 (b - a)} \log |a \cos^{2} x + b \sin^{2} x| + C \end{array} \end{array}$

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