When you do an integration, you should always check it by differentiating.
You have:
[imath]\int \sin^7{x} \text{ dx} = -\frac{1}{8} \cos^8{x} + c\\
\text{ }\\
\text{but }\frac{d}{dx} \left( -\frac{1}{8}\cos^8{x}\right)= -\frac{1}{8} \left( 8\cos^7{x} \sin{x} \right) \text{ not } \sin^7{x}[/imath]
So,in fact, contrary to appearances [imath]\int \sin{x} \cos^7{x} \text{ dx}[/imath] is easy,
[imath]\int\sin^7{x} \text{ dx}[/imath] is harder.
That's why, when I saw your question, noticing that there was an odd number of sines, I kept one and converted the others to cosines, knowing that I would then be able to integrate easily.