∫sin⁡x+cos⁡xsin⁡(x−α)dx is equal to - Infinity Learn by Sri Chaitanya

A
$\begin{array}{l} (\cos α - \sin α) (x - α) + (\cos α + \sin α) \log | \sin (x - α) | + C \end{array}$
B
$\begin{aligned} (\cos α + \sin α) (x - α) + (\cos α - \sin α) \log | \sin (x - α) | + C \end{aligned}$
C
$\begin{aligned} (\cos α + \sin α) (x + α) + (\cos α - \sin α) \log | \sin (x + α) | + C \end{aligned}$
D
none of these

Solution:

We have,

$\int \frac{\sin x + \cos x}{\sin (x - α)} d x$

$= \int \frac{\sin (t + α) + \cos (t + α)}{\sin t} d t,$ Where $t = x - α$

$\begin{array}{l} = \int (\cos α + \sin α \cot t + \cos α \cot t - \sin α) d t \\ = (\cos α - \sin α) t + (\sin α + \cos α) \log | \sin t | + C \\ = (\cos α - \sin α) (x - α) + (\cos α + \sin α) \log | \sin (x - α) | + C \end{array}$

$\int \frac{\sin x + \cos x}{\sin (x - α)} d x$ is equal to

Solution:

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