First slide

Methods of integration

Question

Let $f (x) = \frac{\sqrt{\tan x}}{\sin x \cos x}$ and $F (x)$ is its anti-derivative, if $F (π / 4) = 6$ then $F (x)$ is equal to

Easy

A

$2 (\sqrt{\tan x} + 1)$
B

$2 (\sqrt{\tan x} + 3)$
C

$2 (\sqrt{\tan x} + 2)$
D

none of these

Solution

$F (x) = \int \frac{\sqrt{\tan x}}{\sin x \cos x} d x = \int \frac{\sqrt{\tan x}}{\tan x} \sec^{2} x d x$
$(t = \tan x)$
$= \int \frac{1}{\sqrt{t}} d t + C = 2 \sqrt{t} + C = 2 \sqrt{\tan x} + C$
Since $6 = F (π / 4) = 2 + C so C = 4$ Hence
$F (x) = 2 (\sqrt{\tan x} + 2)$

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