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Evaluate the integral.

$ \displaystyle \int \frac{x^5 + x - 1}{x^3 + 1}\ dx $

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$$\frac{x^{3}}{3}-\ln |x+1|+C$$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Missouri State University

Campbell University

Baylor University

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.

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Evaluate the integral.…

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Evaluate the definite inte…

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Evaluate the integrals.

Let's evaluate the given integral since the numerator has degree larger than the denominator. Let's go ahead and do polynomial division exit the fifth over X cubed. That's ex clear, so it's couldn't multiply that out. Then you subtract and your leftover with negative X squared plus X minus one, and this will be a remainder because X cubed is not going to x squared. So coming back to the original problem, we can rewrite this integral. Our quotient was X squared. That's from up here. And then we had our remainder. So let me go ahead and pull up this minus sign. So we have a minus. Actually, like, I am going to take a step back here. I'm under just right. This is a plus plus our remainder x cubed plus one. Okay. And then now break this into two into girls. And for the second inaugural, let me just plug this minus and from the X Square. So then I'll have X square and then we have to change the sign and the other terms because we factor out a minus sign. Yeah, there's a formula for X Q plus one. This is coming from This is an expression of the form a queue. Plus be cute. We're here Is X B is one. And there's a formula that says this is equal to a plus speed a square minus a B plus peace Where so, using this formula, we have X plus one. But then we have X squared minus X plus one, and we could see that there's a cancellation here. This quadratic terms cancels nicely with the other. So we're just left with in a girl. It's square minus. So she put a DX. They're technically minus in a girl, one over X plus one. Now, for the first interval, we can just use the power rule. That's X cubed over three for the second. And the girl that, plus one is bothering you. Feel free to go. Do you, sir? Here. You could take you to just be f plus one. Then you have natural log. Absolute value X plus one. And then don't forget the constancy of integration. And that's our answer.

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