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Questions  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Integrate $\begin{align*} \int \frac{\sin(x+a)}{\sin (x+b)} dx\end{align*} $


$\begin{align*} (A)\; x \cos (a-b) + \cos (a-b) log |\cos (x+b)| + c \end{align*}$
$\begin{align*} (B)\;x \cos (a-b) + \sin (a-b) log |\sin (x+b)| + c \end{align*}$
$\begin{align*} (C)\; x \sin (a-b) + \cos (a-b) log |\sin (x+b)| + c \end{align*}$
$\begin{align*} (D)\; x \cos (a-b) + \cos (a-b) log |\sin (x+b)| + c \end{align*}$

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$\begin{align*}Let \; I = \int \frac{\sin (x + b + a - b)}{\sin (x+b)} dx\end{align*}$
$\begin{align*}=\int \frac{\sin \{(x+b)+(a-b)\}}{\sin (x+b)} dx \end{align*} $
$\begin{align*} = \int \frac{\sin (x+b) \cos (a-b) + \cos(x+b) \sin(a-b) }{\sin (x+b)} dx \end{align*}$
$\begin{align*} = \int \cos (a-b) + cot (x+b) \sin(a-b) dx \end{align*}$
$=\begin{align*}\cos(a-b) \int dx + \sin (a-b ) \int cot (x+b) dx \end{align*}$
$\begin{align*}= x \cos (a-b) + \sin (a-b) log |\sin (x+b)| + c \end{align*}$
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