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Evaluate the integral.

$ \displaystyle \int_0^1 \frac{x}{(2x + 1)^3}\ 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

Harvey Mudd College

Boston College

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.

06:37

03:17

Evaluate the integral.…

08:47

01:09

06:46

01:07

09:23

01:44

03:24

Evaluate the definite inte…

01:14

01:48

Let's use a U sub for the following. Take you to be two and two plus one. Then here this's equivalents who X equals you minus one over two. And we also have do you over too equals the X for us. Rewrite this one half. Now we have X so that we could use this over here. Then we have our do you We already took out the two over here. And then here we have you. Cute. What? And let's change those limits. If X equals zero, we'LL see you equals one. If that's equals one, then we see U equals three. So that's opera limit. So here, luscious rewrite this. And then we could go ahead and split this into two two fractions and then use the power rule. No plug in your end points. So here, planning the three first? No. Then plug in one. And this should end up simplifying too. One over eighteen. Wait. And that's our answer

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