# How do you find the integral of (sin(5x))(5cos(5x))2?

Jan 31, 2015

The answer is: $- \frac{1}{2} \cos 10 x + c$.

The double-angle formula says:

sin(2alpha)=2sinalphacosalpha) .

So: $2 \sin 5 x \cos 5 x = \sin 10 x$.

Than the integral becomes:

$\int 5 \sin 10 x \mathrm{dx} = \frac{1}{2} \int 10 \sin 10 x \mathrm{dx} = - \frac{1}{2} \cos 10 x + c$,

where I used the integration formula:

$\int \sin f \left(x\right) f ' \left(x\right) \mathrm{dx} = - \cos f \left(x\right) + c$.