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Evaluate the integral.

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

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$I=\left(x^{2}+2 x\right) \sin x+(2 x+2) \cos x-2 \sin x+C$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 1

Integration by Parts

Integration Techniques

Missouri State University

Harvey Mudd College

Baylor University

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 integral.…

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The problem is evaluated the integral x, squared plus 2 x times cosine x d x. For this problem we will use the method of integration by parts, so the formula is integral of upraisequal to u times v minus integral of u prime v d x. For this problem we can let? U is equal to x square plus, 2 x and prim is equal to cosine x. Then, u is equal to 2 x, plus 2 and v is equal to sine x. Now this integral is equal to so this is x, squared plus 2 x times sine x, minus integral of 2 x plus 2 has sine x dx. Now we compute this part for this part, that, u is equal to 2 x as 2 and v from is equal to sine x, then we have primisequal to 2 and v is equal to negative cosine x. This integral is equal to 2 x plus 2 times negative cosine x, minus integral 2 times negative cosine x. This is equal to 2 x plus 2, a negative cosine x, plus 2 sine x. S cast number c, then to whole thing is equal to x, square plus 2 x sine x, so here minus this 1, but this is plus 2 x, plus 2 cosine x, minus 2 sine x and plus some constant number c. This is the answer for the integral of this function.

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