Ask Questions, Get Answers

Want to ask us a question? Click here
Browse Questions
Home  >>  CBSE XII  >>  Math  >>  Integrals
0 votes

Integrate the function $\large\int\frac{\sin x}{\sin(x-a)}$

Can you answer this question?

1 Answer

0 votes
  • (i) if given $I=\int f(x)dx,$ let f(x)=t, then $f'(x)dx=dt$,then
  • (ii)$\sin (A+B)=\sin A \cos B+\cos A\sin B$
  • (iii)$\cot x dx=\log|\sin x|+c$
Step 1:
Given $I=\large\int\frac{\sin x}{\sin(x-a)}$$dx$
Let $x-a=t,$ on integrating we get,
Therefore $x=t+a$
Substituting for t and dt we get,
$I=\large\int\frac{\sin (t+a)}{\sin t}$$dt$
But we know $\sin (A+B)=\sin A \cos B+\cos A \sin B$
Therefore $ I=\large\int \frac{\sin t \cos a+\cos t \sin a}{\sin t}$
Step 2:
On seperating the terms,
$I=\large\int \frac{\sin t \cos a}{\sin t}$$dt+\int\large \frac{\cos t \sin a}{\sin t}$$dt$
$\;\;\;=\int cos a\;dt+\int \sin a. \cot t\;dt$
$=\cos a\int dt+\sin a\int \cot t\;dt$
On integrating we get
$\cos a(t)+\sin a.\log|\sin t|+c$
Substituting for t we get
$\cos a(x-a)+\sin a.\log|\sin (x-a)|+c$
$=x\cos a+\sin a.\log|\sin (x-a)|+a \cos a+c$
Since $a \cos a$ is a constant
$=x\cos a+\sin a.\log|\sin (x-a)|+c$
answered Sep 10, 2013 by sreemathi.v
Ask Question
student study plans
JEE MAIN, CBSE, NEET Mobile and Tablet App
The ultimate mobile app to help you crack your examinations
Get the Android App