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Evaluate the integral.

$ \displaystyle \int (x + \sin x)^2\ dx $

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Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 5

Strategy for Integration

Integration Techniques

Campbell University

Harvey Mudd College

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.

01:55

Evaluate the integral.…

04:15

Evaluate the indefini…

02:14

02:33

00:37

Evaluate the indefinite in…

00:39

00:33

02:03

Let's start off by just expanding this into Grand X Square to Excite X Science, where then we could go ahead and split this sense of three generals. Oops, think of the ex first and then in a gruel of science squares. So for the first one, just use the power room here. Let's use and enrich my parts. Let's take you to be x two u equals the x t me signed the eggs. So he is negative. Cosign x sound. Recall the formula to be replaced the integral with UV minus in rule be, do you? So we have plus and then two times. So now we do you times V So this times this that's negative. X co sign X and then minus in a rule z Do you? So I have a negative co sign. Let me just go ahead and cancel those minuses. And then finally, for the last integral. It's used a half angle formula for sign, so it's one minus co sign to X, and we pulled out the two. So let's simplify this integral of co sign a sign. Don't forget the two out here. So here, in a goal of one X. Don't forget the one half hour in the front, out and then the end. Roll over here that'LL be a minus sign, too. It's over, too. But then we have another two over here. So as a four plus our constant of integration, see? And that's your final answer.

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