What is the integral of #int sin^4(x) dx#?
This integral is mostly about clever rewriting of your functions. As a rule of thumb, if the power is even, we use the double angle formula. The double angle formula says:
If we split up our integral like this,
We can use the double angle formula twice:
Both parts are the same, so we can just put it as a square:
Expanding, we get:
We can then use the other double angle formula
to rewrite the last term as follows:
I will call the left integral in the parenthesis Integral 1, and the right on Integral 2.
Looking at the integral, we have the derivative of the inside,
If we let
It's not as obvious here, but we can also use u-substitution here. We can let
Completing the original integral
Now that we know Integral 1 and Integral 2, we can plug them back into our original expression to get the final answer: