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Evaluate the integral.

$ \displaystyle \int \sin 2 \theta \sin 6 \theta d \theta $

$$

\frac{1}{8} \sin 4 \theta-\frac{1}{16} \sin 8 \theta+C

$$

Integration Techniques

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Missouri State University

Campbell University

Idaho State University

This problem is from chapter seven, Section two. Problem number forty two in the book Calculus Early. Transcendental. Lt's a Tradition by James Door. We have an indefinite general sign of Tooth Taylor time Sign of six data. So are integral. Our instagram is of the form sign a sign be in our case we have is to data will be a six data so we can rewrite the Inter Grand using this formula here. And the purpose for doing this is that the right hand side will be easier to integrate because the co signs in the signs on the right hand side are not being multiplied together. So is pull out that one half outside the inner girl. We have co sign of two. They don't minus six data minus co Sign of to theatre plus six data. So separate us from our work on the side. So here we can observe that we have to fade on minus six data. So they're sort here is negative forthe Daito. And if we like, we can replace co signer minus for data with co sign of fourth Ada. Since you may recall, that coastline is even not necessary. But it'LL Simplify I work. We have one has integral co signing for data minus cosign a beta Dana. And here in my help to go ahead and apply a u substitution. If you're unsure about these animals here for this one, you could take you to be four Daito. And for this integral, you can take you to be a data. So evaluating these two minute girls, we should obtain sign Fourth eight over four minus sign eight data over eight, plus our constancy. And this could be simplified a little bit. So let's go ahead and do that sign for theta over eight minus. Sign a beta over sixteen. Plus he and there's our answer.