OK so when we're evaluating integrals of the form

$\displaystyle \int(sin^n x)dx$

we know reduction formulas from the past that make it simpler to evaluate them which is

$\displaystyle \int \sin^n x dx = - \frac{\ \sin^{n-1} x \cos x}{n} + \frac{\ n-1}{n} \int \sin^{n-2} x dx$

however my problem is when the term inside the $\displaystyle sin $ isn't just an x

In my case, it's

$\displaystyle \int sin ^2 (5x)dx$

What do i do? I'm pretty sure it changes something because I'm off by a factor. My final answer is

$\displaystyle -\frac{1}{4}sin(10x) + \frac{x}{2} + C $

However the correct answer is

$\displaystyle -\frac{1}{20}sin(10x) + \frac{x}{2} + C$

I'm off by a factor of 5. Coincidence?