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Problem

Evaluate the indefinite integral. Illustrate, and…

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Problem 53 Hard Difficulty

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking $ C = 0 $).

$ \displaystyle \int \sin 3x \sin 6x dx $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 2

Trigonometric Integrals

Related Topics

Integration Techniques

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Video Thumbnail

01:53

Integration Techniques - Intro

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.

Video Thumbnail

27:53

Basic Techniques

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

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Watch More Solved Questions in Chapter 7

Problem 1
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Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
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Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
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Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
Problem 35
Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
Problem 44
Problem 45
Problem 46
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Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
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Problem 69
Problem 70

Video Transcript

no For this problem will find the anti derivative of signed three X time science XX Then we'LL check. Their answer is reasonable by first graphing the interbrand which is circled in red as well as the entire anti derivative. So first, let's just go ahead and find the anti derivatives. So one way to proceed is to go ahead and re write us. I noticed that six X is two times three x So I take the second determine here and I write is sign of two times three x. So this is corresponding to the science six X Here I did was rewrite this No into this. And then from this new term sign of two times three x, we can use this double angle formula to rewrite this. So we have the integral signed three eggs. Times go different color here, So we have to signed three eggs. Times co signed three X. So here we're using T equals three X and our problem. Okay, so then we could go ahead and simplify than integrate. So here is pulled. That too. We have a sine squared three x cosign three x and then this looks like we should do it. Use up. Let's take you two be signed three x then do you is coastline three eggs times three by the general So that do you over three is co signed three x So here we have to theirs in a girl You square Do you which is to you cute over nine plus c by the instructions were supposed to take seed of zero so we could go in and drop this constant here and then lastly, we can go back to our U substitution toe back substitute. So here we have to over nine Sign Cube, three x. So this is our final answer. This is our anti derivatives. So let's call this in blue. No. So and again we've coordinated the instagram with red. So if the integral of the red function equals the blue function, that means that the derivative of the blue function so derivative of the blue equals the red. Then we can go to the graphing calculator toe check. If this is reasonable, so score the gravity calculator. So the red graph is supposed to be the derivative of the blue graph. So let's first check to see when the red graph should be zero. So let's see, when the blue graph has a horizontal tangent. Here at the origin, we see a horizontal tension that means the rivet of a zero and the Red graphic zero. Here, at about point five, we see a horizontal tangent on the blue. That means that the riveting zero and that's where the red is. And same over here. At about one point one, we see a horizontal changes on the blue and the road graphic zero, and you could go out and check that every time. The blue graph has a horizontal tension, the red graph zero and maybe from to convince you further hear from zero to about point five, we see that the blue graph is increasing. That means that the derivative is positive. So the red graph should be above the X axis from zero point five, and we see that's exactly what's happening. So these conclusions suggest that our answer's correct

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Related Topics

Integration Techniques

Top Calculus 2 / BC Educators
Catherine Ross

Missouri State University

Anna Marie Vagnozzi

Campbell University

Kayleah Tsai

Harvey Mudd College

Caleb Elmore

Baylor University

Calculus 2 / BC Courses

Lectures

Video Thumbnail

01:53

Integration Techniques - Intro

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.

Video Thumbnail

27:53

Basic Techniques

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

Join Course
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