First slide

Methods of integration

Question

Let $f (x) = \frac{1}{4 - 3 \cos^{2} x + 5 \sin^{2} x}$ and if its antiderivative $F (x) = (1 / 3) \tan^{- 1} (g (x)) + C$ then $g (x)$ is
equal to

Easy

A

$3 \tan x$
B

$(\sqrt{2}) \tan x$
C

$2 \tan x$
D

none of these

Solution

$\begin{array}{l} F (x) = \int \frac{d x}{4 - 3 \cos^{2} x + 5 \sin^{2} x} + C \\ = \int \frac{\sec^{2} x d x}{4 (1 + \tan^{2} x) - 3 + 5 \tan^{2} x} + C \\ = \int \frac{d t}{9 t^{2} + 1} + C \\ = \frac{1}{3} \tan^{- 1} 3 t + C \end{array} (t = \tan x)$
Therefore $g (x) = 3 \tan x$

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