Question

What is the integral of `sin(x)*sin(10x)`?

Answer 1

Use one of the product to sum formulas. Not many of us use them enough to memorize them, but they are easy enough to get.

We want to change a product of sines. We recall the sine and cosine of a sum or difference.

For this problem we'll need

`cos(a+b)=cosacosb-sinasinb`
`cos(a-b)=cosacosb+sinasinb`

Subtracting the first equation fron the second gives us:

`cos(a-b)-cos(a+b)=2sinasinb`

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

In this problem `a=x` and `b=10x`

`sinxsin10x=1/2cos(-9x)-1/2 cos11x`

Since cosine is an even function, make it
`1/2 cos9x - 1/2cos11x`

Now integrate each term by simple substitution.

Notes

To change a product of cosines, add the two formulas above.
`cosacosb= 1/2 cos(a-b) + 1/2 cos(a+b)`

To change `sinacosb`, write the formulas for `sin(a+b)` and `sin(a-b)` and add.