First slide

Methods of integration

Question

Let $f (x) = \frac{1}{1 - \sin^{4} x}$ and $F$ be an antiderivative
of $f$ .
Statement-1: $F (x) = \frac{1}{2} \tan x + \frac{1}{2 \sqrt{2}} \tan^{- 1} (\sqrt{2} \tan x) + C$
Statement-2: $F$ is a one-one function of $\tan x$ .

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

$f (x) = \frac{1}{\cos^{2} x (1 + \sin^{2} x)} = \frac{\sec^{2} x (1 + \tan^{2} x)}{1 + 2 \tan^{2} x}$
So, $\begin{aligned} F (x) = \int \frac{1 + t^{2}}{1 + 2 t^{2}} d t (t = \tan x) \\ = \int (\frac{1}{2} + \frac{1}{4} \frac{1}{t^{2} + 1 / 2}) d t \end{aligned}$
$\begin{array}{l} = \frac{1}{2} t + \frac{1}{4} \sqrt{2} \tan^{- 1} (\sqrt{2} \tan x) + C \\ = \frac{1}{2} \tan x + \frac{1}{2 \sqrt{2}} \tan^{- 1} (\sqrt{2} \tan x) + C \end{array}$
which is a 1 – 1 function of $\tan x$ .

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