Question

I want to know how to integrate Sin(2x)*Sin(5x)dx?

Answer 1

`I=1/42(7sin(3x)-3sin(7x))+C`

Explanation:

We want to solve

`I=intsin(2x)sin(5x)dx`

Use the product identity

`sin(a)sin(b)=1/2(cos(a-b)-cos(a+b))`

With `a=5x` and `b=2x`

`I=1/2int(cos(3x)-cos(7x))dx`

Which is much more comfortable to integrate

`I=1/2(1/3sin(3x)-1/7sin(7x))+C`

Or

`I=1/42(7sin(3x)-3sin(7x))+C`