Solving integrals of the form sinm (x) cosn (x)

By Tutor GuyNo Comments


An important class of integrals is of the form:

\int \sin^m{x} \; \cos^n{x} \; dx

where m and n are integers. If either m or n is odd, factor out a single power of that function and rewrite the integral to solve with a u substitution. This is best demonstrated with an example:

\int \sin^5{x} \; \cos^7{x} \; dx

Here, both m and n are odd, so we can select either function to factor. The process is easier when you pick the smaller of the two exponents, so let’s choose the sin x and we factor out one power as follows:

\int \sin{x} \; \sin^4{x} \; \cos^7{x} \; dx

We then rewrite the even power as a power of sin2 x so that we can apply a trig identity:

\int \sin{x} \; \sin^4{x} \; \cos^7{x} \; dx = \int \sin{x} \; (\sin^2{x})^2 \; \cos^7{x} \; dx =

\int \sin{x} \; (1-\cos^2{x})^2 \; \cos^7{x} \; dx = \int \sin{x} \; (1-2 \cos^2{x} + \cos^4{x}) \; \cos^7{x} \; dx =

\int \sin{x} \; (\cos^7{x} -2 \cos^9{x} + \cos^{11}{x}) \; dx

This is easily integrated with a u substitution (let u = cos x):

\dfrac{1}{8} \cos^8{x} - \dfrac{1}{5} \cos^{10}{x} + \dfrac{1}{12} \cos^{12}{x} + C

So what do you do if both m and n are even? Well, most students just skip the problem and go on to the next one. ☺ But if you want to see the technique, look for my next blog post.

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