Integrate the function in Exercise: (x-a) - Vidyadip

Question

Integrate the function in Exercise:
$\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}$

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Answer

$\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}$
$\text{Let}\ \text{x}-\text{a}=\text{t}\Rightarrow\text{dx}=\text{dt}$
$\int\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}\text{dx}=\int\frac{\sin(\text{t}+\text{a})}{\sin\text{t}}\text{dt}$
$=\int\frac{\sin\text{t}\cos\text{a}+\cos\text{t}\sin\text{a}}{\sin\text{t}}\text{dt}$
$=\int(\cos\text{a}+\cot\text{t}\sin\text{a)}\text{dt}$
$=\text{t}\cos\text{a}+\sin\text{a}\log|\sin\text{t|}+\text{C}_{1}$
$=(\text{x}-\text{a)}\cos\text{a}+\sin\text{a}\log|\sin(\text{x}-\text{a)}|+\text{C}_{1}$
$=\text{x}\cos\text{a}+\sin\text{a}\log|\sin(\text{x}-\text{a)}|-\text{a}\cos\text{a}+\text{C}_{1}$
$=\sin\text{a}\log|\sin(\text{x}-\text{a)}|+\text{x}\cos\text{a}+\text{C}$

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