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Problem

If $ a \neq 0 $ and $ n $ is a positive integer, …

04:37

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Problem 74 Hard Difficulty

If $ f $ is a quadratic function such that $ f(0) = 1 $ and
$$ \int \frac{f(x)}{x^2 (x + 1)^3}\ dx $$
is a rational function, find the value of $ f^\prime (0) $.


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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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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.

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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
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
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Problem 65
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Problem 68
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Problem 71
Problem 72
Problem 73
Problem 74
Problem 75

Video Transcript

F is a quadratic function. F zero was one and were given that this in a girl's irrational function and would like to find at prime of zero. So let's come back up here and use this information. We have zero equal C equals one. So that gives us see So we have a of X equals x, where, plus b X plus one. And the thing that we're after is a primal zero. So that's just need. So we'd like to find the value of be such that this thing is rational function, so that will determine the possible values for me. So let's go ahead and do partial fraction to composition. So right now our instagram is f of X, which we know is that up there and then we have X squared. That's plus one cute. Now, using what the author calls case, too, I can write. This is ale rex be over ex players and then it using case to again, this time for X plus one she over X plus one deal Rex plus one square e over at plus one. Cute Oops And then let's go ahead and most supply Well, actually, at this case was before we multiply, we may not have to hear. Let's just note that we want this intervals to be irrational function. So, in particular, rational means we don't want logarithms No lots here, so that automatically eliminates these two terms here. We do not want these terms because these will give us natural logs. So this gives us a and sear both zero. So when I go ahead and multiply both sides by this denominator and we're getting rid of A and C score the next page, I get a X Square, be explicit one and then on the right, and then we could go ahead and solve this. So go ahead and expand the sow and then combine based on the power of ex supporter X cubed pullout of X squared. And that's what we're doing right now. So that's when you pull out X Square and then three b x. Well, then the and then we end up with three. B equals little bee. Large B is one, so that gives us B equals three. And that's exactly what we wanted. Because remember from the previous page, we wanted a crime of zero. We knew that was deep the be over here in front of the X, and we just found that value will be that gives us three, and that's your final answer.

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Related Topics

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Top Calculus 2 / BC Educators
Grace He

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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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