Reduction formula for Integral of ∫sin^n(x)dx

Let's consider the integral of the n-th power of the sine function, sinⁿ(x), where n = 1, 2, 3,... (natural numbers). Since sine is a function of x, we can't simply apply the standard "power" formula like we do with powers of x on its own.

So how do we go about the following problem?

Well, in the "real" world of real functions, there really are only 2 methods of integration:

1. Integration by substitution
2. Integration by parts

We'll tackle this problem with integration by parts.

To do this, we'll need to "breakup" the integrand sinⁿ(x) into 2 parts. We can write:

...and we can integrate both sides...

Now, if we remember the formula for integration by parts...

...we can let:

...and:

So, to get du and v, we need to differentiate u and integrate dv. Thus, first applying the chain rule to u to get du/dx:

...and integrating dv:

So now that we have all of our parts, let's apply the integration by parts formula:

Now, for the trailing integral on the right-hand side, the integrand contains a cosine-squared term. We can apply the Pythagorean Identity to rewrite this integral as follows...

Now, putting the above into the context of our problem, we have...

We can then move the 2nd integral on the right-hand side over to the left-hand side to complete the problem...

And finally, dividing both sides by n...

So there we have it! The reduction formula for the integral of the n-th power of the sine function. Now, whilst this formula is valid for any value of integer n greater than or equal to 1 (i.e. natural numbers), it is best to use this for powers greater than 3, as n = 1, 2, 3 have strategies for integration.

If you need a clearer explanation, play my video below...

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