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Use a graph of $ f(x) = \frac{1}{(x^2 - 2x - 3)} …

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Problem 54 Medium Difficulty

Use integration by parts, together with the techniques of this section, to evaluate the integral.

$ \displaystyle \int x \tan^{-1} x\ dx $


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Related Topics

Integration Techniques

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Top Calculus 2 / BC Educators
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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 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
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
Problem 29
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
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
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Problem 72
Problem 73
Problem 74
Problem 75

Video Transcript

let's use integration by parts, possibly with the techniques of the section. We may have to do partial fractions after immigration My parts well issues. This is our starting point here and maybe easiest to just go ahead and take you to be our camp. We know the derivative of that one over X squared plus one D x devi is X tx So then the expert over too. So you think in aggression my parts recall that this is UV minus in roll video. So for you, times v ex cleared over two times are tan and then minus in a girl And then we have VD you. Let's go ahead and pull that one half are the rules. Now before we try partial fraction to composition if it's even necessary First we should do long division here because the numerator has degree That's equal. So the denominator Saagar decide to do that and we have a remainder of minus one. This means we can rewrite this all. So here we have our quotient which was one and there are remainder was minus one. So we have minus one over X squared plus one. Now we know from basically going backwards from to you to you, We know that the integral of one over X squared plus one is our ten. So otherwise, if you have forgotten that factor didn't see this up here, you could go ahead and do it shrinks up if you had to. And that will give you the anti derivative. So now let's evaluate X squared over to our ten and then we have minus one half and then x minus Ark ten of X. We've just evaluated the integral. So don't forget to add that constant of Integration City. And then the last step is maybe we can go ahead and just distribute this net negative one half. So I'll go to the next page to write that X squared over to ten in verse of X. This was our UV term, and then the general gave us this and that's your final answer

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Top Calculus 2 / BC Educators
Anna Marie Vagnozzi

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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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