4. Describe the method used to integrate sinᵐx cosⁿx, for m ev...

Question

4. Describe the method used to integrate sinᵐx cosⁿx, for m even and n odd.

Accepted Answer

Welcome back, everyone. Given the integral of sin to the power of 6 of X, cosine cubed of XDX, which substitution is most effective after expressing all but one cosine X in terms of sine X? So let's consider our integral. It is integral of sin to the power of 6 of X, cosine cubed of XD X, and as the problem suggests, we're going to essentially split our cosine cubed, right? So we're going to rewrite it as the integral of sin to the power of 6 of X. Cosine. Now, our cube is essentially the product between a square and a terms of the power of 1, right? So that'd be cosine squared of X multiplied by cosine X. So here we are isolating cosine to the power of 1, and now let's add the X. So now what we can do is simply use the Pythagorean identity. We're going to leave to the power of 6 of X, and now cosine square x can be rewritten as 1 minus square x using the Pythagorean identity. And this is multiplied by cosine xDX. And now let's focus on cosine xDX. Let's recall that cosine x is the derivative of sine. So what we can do is simply set U equals sine x because the derivative DU divided by D X. Is going to be equal to cosine x and rearranging this expression, we get the U equals cosine xDX, right? So, cosine XDX is going to become DU. Sin to the power of 6 of X is going to become a U to the power of 6 and 1 minus square of X is going to become 1 minus U2. So the overall integral can now be written as U to the power of 6 multiplied by. 1 minus. U squared D U. And then we can easily integrate it, right, using the power rule if we distribute. So that's integral or if you use the power of 6 minus, you use the power of 8DU. From here we can apply the power rule. It is not necessary for this problem to evaluate the integral, but now we can see that using the power rule we can get our answer, meaning the correct answer for this problem would be U equals sin x. This is the most effective substitution. That allows us to simplify the integral. Thank you for watching.

4. Describe the method used to integrate sinᵐx cosⁿx, for m even and n odd.

Key Concepts

Integration by Parts

Trigonometric Identities

Reduction Formulas

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