integration - How can I find $int(sin ^4 x ) dx$?

Hi,

My question is: How can I solve the following integral question?

$$\int(\sin ^4 x ) dx$$

Thanks in advance.

Answer

There are standard techniques for solving integrals that consist of powers of sines and cosines.

One is to use reduction formulas when you simply have a power of sines or a power of cosines. These can be obtained by performing integration by parts, followed by using a trigonometric identity. This is done in pretty much every single calculus textbook I have ever encountered.

Another is to use trigonometric power reduction formulas to change the fourth power of the sine into a sum of multiples of simple cosines, which can then be solved with an easy substitution.