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Evaluate the integral.

$ \displaystyle \int \frac{1}{x^2 \sqrt{4x + 1}}\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Campbell University

Oregon State University

University of Michigan - Ann Arbor

University of Nottingham

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 definite inte…

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Evaluate the integral.…

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Let's try you sub here Let's take you to be for X plus one in the radical Then do you two over the radical and we can write This is two over you Do you? Excuse me? DX and then go ahead and solve this for D X. Okay, Now let's go to the original So we have just the one on top. Then we have the X. So that'LL be you over too Then here we see that in you But we also have X squared So to get that to your ex square bull sides and in sulfur x What? And then go ahead and square that to get X squared So we get you square minus one over four and we square the whole thing So this will be our explorer now we'LL see if we can cancel We could cancel these use And then we can write this four square but in the numerator. And then when we divide that by two you get a So we have one over you swear minus one square And then let's go to the next page. God in factor. Oops! Now here we would need to do partial fractions so we eh? You plus one d up D up and see a top. Okay. And then he ends up. So here you will find a, B, C and D, and we get the following one fourth for a also for B one over four. What? And then for C minus one over four. But then for the positive one over four. Now, just go ahead and use the power rule here for all these. So we have to natural log you plus one and then minus two, also minus two here. And then one more mind was too. And then our constancy. And then here we could go back in terms of x. So recall. Yeah, that was our use of So it's gotten plug this in for X for you. So it's you plus one minus two over you plus one and then minus two natural long. And then here we have you minus one Well, and then minus to you. And that's our answer.

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