The value of ∫sin⁡x+cos⁡x3+sin⁡2xdx, is - Infinity Learn by Sri Chaitanya

A
$\frac{1}{4} \log (\frac{2 + \sin x - \cos x}{2 - \sin x + \cos x}) + C$
B
$\frac{1}{2} \log (\frac{2 + \sin x}{2 - \sin x}) + C$
C
$\frac{1}{4} \log (\frac{1 + \sin x}{1 - \sin x}) + C$
D
none of these

Solution:

We have,

$\begin{array}{l} I = \int \frac{\sin x + \cos x}{3 + \sin 2 x} d x \\ \Rightarrow I = \int \frac{1}{4 - (1 - \sin 2 x)} d (\sin x - \cos x) \\ \Rightarrow I = \int \frac{1}{2^{2} - (\sin x - \cos x)^{2}} d (\sin x - \cos x) \\ \Rightarrow I = \frac{1}{4} \log (\frac{2 + \sin x - \cos x}{2 - \sin x + \cos x}) + C \end{array}$

The value of $\int \frac{\sin x + \cos x}{3 + \sin 2 x} d x,$ is

Solution:

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