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Evaluate the integral.

$ \displaystyle \int \ln (1 + x^2)\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

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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01:54

Evaluate the integral.…

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00:33

Evaluate the indefinite in…

01:52

Let's use integration by parts for this integral here. Let's just take you to just be the intern grand. So then go ahead and use the chain rule to differentiate us. So you get one over one plus x Square, but then take the derivative of one plus x squared. So that's our do you Then we're left over with TV equals the X. So vehicles X now using integration by parts here, recalled the formula U V minus and enroll video. So we'LL go ahead and plug in our use. Andy's here, so we have X natural log one plus X squared, so lets you times v and then we have minus in a girl VDO. So here we have the and then do you So let me go in and pull out that, too. And then X squared over one plus x squared. Now you may try partial fractions here, but first we see that they have the same degree. So here we would need to do polynomial division and we can go ahead and rewrite this. After doing the long division, we get one minus one over one plus x cleared. Yeah, and that is our partial fraction to composition, So there's no need to find the baby in the seas. And, well, we know how to integrate both of these. If the second one. If this if you don't remember what this one is, you could do a trousseau at sequel stand data. So after writing this, we have X ellen one plus X square minus two in a girl, one minus one over one plus x squared DX. So now X Ellen one plus x square. Then from this we have a minus two X when we integrate And then after doing the tricks up here, you'LL get a plus two times ten inverse x and then we'LL add our constant of integration, see? And that's your final answer.

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