The value of ∫sec⁡xdxsin⁡(2x+θ)+sin⁡θ is - Infinity Learn by Sri Chaitanya

A
$\sqrt{(\tan x + \tan θ) \sec θ} + C$
B
$\sqrt{2 (\tan x + \tan θ) \sec θ} + C$
C
$\sqrt{2 (\sin x + \tan θ) \sec θ} + C$
D
none of these

Solution:

The given integrals can be written as

$\begin{array}{l} \int \frac{\sec x d x}{\sqrt{2 \sin (x + θ) \cos x}} \\ = \frac{1}{\sqrt{2}} \int \frac{\sec^{3 / 2} x d x}{\sqrt{\sin x \cos θ + \sin θ \cos x}} \\ = \frac{1}{\sqrt{2}} \int \frac{\sec^{2} x d x}{\sqrt{\tan x \cos θ + \sin θ}} \\ = \frac{1}{\sqrt{2}} \int \frac{d t}{\sqrt{t \cos θ + \sin θ}} \\ = \frac{2}{\sqrt{2}} \sqrt{t \cos θ + \sin θ} + C \\ = \sqrt{2 (\tan x \sec θ + \tan θ \sec θ)} + C \end{array}$

The value of $\int \frac{\sec x d x}{\sqrt{\sin (2 x + θ) + \sin θ}}$ is

Solution:

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