Please help me evaluate: $ \int\frac{dx}{\sin(x+a)\sin(x+b)} $
Evaluate $\int\frac{dx}{\sin(x+a)\sin(x+b)}$
5
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calculus
integration
2 Answers
4
The given integral is: $\int\frac{dx}{\sin(x+a)\sin(x+b)}$
The given integral can write:
$\int\frac{dx}{\sin(x+a)\sin(x+b)}=\int\frac{\sin(x+a)}{\sin(x+b)}\cdot\frac{dx}{\sin^2(x+a)}$
We substition $\frac{\sin(x+a)}{\sin(x+b)}=t$
By the substition of the above have:
$\frac{dx}{\sin^2(x+a)}=\frac{dt}{\sin(a-b)}$
Now have:
$\int\frac{dx}{\sin(x+a)\sin(x+b)}=\int\frac{\sin(x+a)}{\sin(x+b)}\cdot\frac{dx}{\sin^2(x+a)}=\frac{1}{\sin(a-b)}\int\frac{dt}{t}=\frac{1}{\sin(a-b)}\ln |t|=\frac{1}{\sin(a-b)}\ln|\frac{\sin(x+a)}{\sin(x+b)}|+C$
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4This is supposed to be a homework question, so I presume you should leave the proposer with some room to do it himself... – 2012-09-04
6
Hint: Multiply and divide by $\sin(b-a)$.
Further Hint:
Write $\sin(b-a) = \sin((x-a)-(x-b))$