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CalculusCalculus (MindTap Course List)8th Edition

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Chapter 7.1, Problem 7E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Solutions

Chapter
Section
Problem 1E
Problem 2E
Problem 3E
Problem 4E
Problem 5E
Problem 6E
Problem 7E
Problem 8E
Problem 9E
Problem 10E
Problem 11E
Problem 12E
Problem 13E
Problem 14E
Problem 15E
Problem 16E
Problem 17E
Problem 18E
Problem 19E
Problem 20E
Problem 21E
Problem 22E
Problem 23E
Problem 24E
Problem 25E
Problem 26E
Problem 27E
Problem 28E
Problem 29E
Problem 30E
Problem 31E
Problem 32E
Problem 33E
Problem 34E
Problem 35E
Problem 36E
Problem 37E
Problem 38E
Problem 39E
Problem 40E
Problem 41E
Problem 42E
Problem 43E
Problem 44E
Problem 45E
Problem 46E
Problem 47E
Problem 48E
Problem 49E
Problem 50E
Problem 51E
Problem 52E
Problem 53E
Problem 54E
Problem 55E
Problem 56E
Problem 57E
Problem 58E
Problem 59E
Problem 60E
Problem 61E
Problem 62E
Problem 63E
Problem 64E
Problem 65E
Problem 66E
Problem 67E
Problem 68E
Problem 69E
Problem 70E
Problem 71E
Problem 72E
Problem 73E
Problem 74E
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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

Evaluate the integral.

∫ ( x 2 + 2 x ) cos x   d x

To determine

To evaluate: The given integral using the technique of integration by parts.

Explanation

The technique of integration by parts comes in handy when the integrand involves product of two functions. It can be thought of as a rule corresponding to the product rule in differentiation.

Formula used:

The formula for integration by parts in terms of u and v is given by

∫udv=uv−∫vdu

Given:

The integral, ∫(x2+2x)cosxdx.

Calculation:

Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=x2+2x     dv=cosxdx

Then, the differentiation of u and antiderivative of dv will be

du=(2x+2)dx     v=sinx

Substitute for variables in the formula above to get

∫(x2+2x)cosxdx=(x2+2x)(sinx)−∫(sinx)(2x+2)dx

Integrate the last term

∫(x2+2x)cosxdx=(x2+2x)(sinx)−∫(2xsinx+2sinx)dx=(x2+2x)(sinx)−∫(2sinx)dx−∫(2xsinx)dx=(x2+2x)(sinx)−2∫sinxdx−2∫xsinxdx

=(x2+2x)(sinx)−2(−cosx)−2C1−2∫xsinxdx …… (1)

Notice that the last term is against an integral of product of two functions

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Finding Arc Length In Exercises 7-20, find the arc length of the graph of the function over the indicated inter...

Calculus (MindTap Course List)

Finding the Equation of a SphereIn Exercises 4346, complete the square to write the equation of the sphere in s...

Multivariable Calculus

Finding the Projection of u onto v In Exercises 37-44, (a) find the projection of u onto v and (b) find the vec...

Calculus

Indicate which of the following situations involve sampling from a finite population and which involve sampling...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

For Exercises 1 through 6, compute angles to the nearer minute or hundredth degree. Three views of a part are s...

Mathematics For Machine Technology

Consider babies born in the normal range of 3743 weeks gestational age. The paper referenced in Example 7.27 Bi...

Introduction To Statistics And Data Analysis

The Transactional Records Access Clearinghouse at Syracuse University reported data showing the odds of an Inte...

Statistics for Business & Economics, Revised (MindTap Course List)

If 0 is a zero of P(x)=anxn+an1xn1+...+a1x+a0. Find a0.

College Algebra (MindTap Course List)

Define plagiarism and explain the techniques that can be used to help avoid unintentional plagiarism.

Research Methods for the Behavioral Sciences (MindTap Course List)

Identify and explain the common threats to external validity and identify threats when they appear in a researc...

Research Methods for the Behavioral Sciences (MindTap Course List)

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