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Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral. $ \displaystyle \int \sin^2 x \cos x \ln (\sin x)\ dx $

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$\frac{1}{9} \sin ^{3} x[3 \ln (\sin x)-1]+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 6

Integration Using Tables and Computer Algebra Systems

Integration Techniques

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

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.

03:14

Use the Table of Integral…

03:28

Use the Table of Integrals…

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03:56

02:17

03:35

07:03

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02:00

02:43

All right, This question wants us to find the value of this integral. So to do this, we need to get it into one of the forms that our table needs. So we need to find a choice of you. So since we see a sign of ex inside of that natural log, that's probably a good guess. So look you equal to sign X. So that means set d'you is equal to co sign x d. X on we see, we do have a co sign there. Indeed. So are integral. Now becomes Thean Tha Gral of well, co sign next D X becomes d'you and then sign squared in front becomes you squared times the natural log of you, Do you now? This is something that are integral. Table can tell us what the value is. So it says, I'll just put this over here. The integral of you to the end. Ellen of you. Do you based on our formula is U to the n plus one over n plus one squared times n plus one Ellen of you minus one plus c so we can see that our inter girls just the n equals two case of this. So this is equal to you to the n plus one, which is three over and plus one squared times the natural log of n plus one times natural log of you minus one plus c. So then, if we look here, we get you cute. Over nine times three Ellen of you minus one. And now all we have to do is plug in. Are you value of sine X? So we get signed cube decks over nine times three natural log sine X minus one plus c and that's our final answer.

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