First slide

Introduction to integration

Question

The function $f$ whose graph passes through (0 ,0 )
and whose derivative is $\frac{\cos^{4} x + \sin^{4} x}{\cos^{2} x - \sin^{2} x}$ is given by

Easy

A

$\frac{1}{4} \log |\frac{1 + \tan x}{1 - \tan x}| + \frac{1}{2} \sin x \cos x$
B

$\log |\frac{\cos x + \sin x}{\cos x - \sin x}| + \sin 2 x$
C

$\log |\frac{\cos x - \sin x}{\cos x + \sin x}| + \frac{1}{2} \sin x \cos x + x$
D

none of these

Solution

We have
$f (x) = \int \frac{\cos^{4} x + \sin^{4} x}{\cos^{2} x - \sin^{2} x} d x$
Using, $2 (\cos^{4} x + \sin^{4} x)$
$= {(\cos^{2} x + \sin^{2} x)}^{2} + {(\cos^{2} x - \sin^{2} x)}^{2}$ , we can write
$f (x) = \frac{1}{2} \int \frac{1}{\cos^{2} x - \sin^{2} x} d x$
$+ \frac{1}{2} \int (\cos^{2} x - \sin^{2} x) d x$
$= \frac{1}{2} \int \sec 2 x d x + \frac{1}{2} \int \cos 2 x d x$
$= \frac{1}{4} \log |\tan (x + \frac{π}{4})| + \frac{1}{4} \sin 2 x + C$
$= \frac{1}{4} \log |\frac{1 + \tan x}{1 - \tan x}| + \frac{1}{2} \sin x \cos x + C$
Since $f (0) = 0$ , we get $C = 0$ .

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App