If f(x)=1cos2⁡x1+tan⁡x then its anti-derivative F(x) ,F(0) = 4 is

A
$\sqrt{1 + \tan x} + 4$
B
$\frac{2}{3} (1 + \tan x)^{3 / 2}$
C
$2 (\sqrt{1 + \tan x} + 1)$
D
none of these

Solution:

$\begin{array}{l} F (x) = \int \frac{d x}{\cos^{2} x \sqrt{1 + \tan x}} + C \\ = \int \frac{1}{\sqrt{1 + t}} d t + C (t = \tan x) \\ = 2 (\sqrt{1 + t}) + C \\ = 2 \sqrt{1 + \tan x} + C \end{array}$

Since $4 = F (0) = 2 + C \Rightarrow C = 2$ Hence

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

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