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Evaluate the integral.

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

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Integration Techniques

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University of Michigan - Ann Arbor

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Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

01:25

Evaluate the following int…

05:51

02:02

01:41

Evaluate the integral.…

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.

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