TITLE: Which is easier to integrate?
QUESTION [5 upvotes]: My calculus teacher gave us this problem in class:
Which is easier to integrate?
$$\int \sin^{100}x\cos x dx$$
or
$$\int \sin^{50}xdx$$
By easier, I assume the teacher means which integral would take less work. I'm unsure of how to approach this problem because of the relatively large exponents. I would guess the second because it has smaller exponents but I'm not sure.

REPLY [2 votes]: Which would you prefer to do?
Let $u=\sin x$ and $du=\cos  x$, transforming the integral into $\displaystyle \int  u^{100} du$.
Or use the reduction formula: $\displaystyle\int \sin^n x dx=\frac{-\sin^{n-1}x \cos x}{n}+\frac{n-1}{n}\int \sin^{n-2}x dx$ for $n=50$? I found the formula here. 
Of course, both can be done, but the reduction formula is incredibly tedious to use. The power rule of integration is much simpler.