   Chapter 5.2, Problem 45E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Comparing Methods In Exercises 43-46, (a) perform the integration in two ways: once using the Simple Power Rule and once using the General Power Rule. (b) Explain the difference in the results. (c) Which method do you prefer? Explain your reasoning. ∫ x ( x 2  + 2 ) 2   d x

(a)

To determine

To calculate: The value of the provided indefinite integral x(x2+2)2dx using simple power rule and general power rule.

Explanation

Given Information:

The provided indefinite integral is x(x2+2)2dx.

Formula Used:

According to the general power rule for integration,

If u is a differentiable function of x, then

undu=un+1n+1+C,

where n1

The algebraic identity of square of sum of two real numbers is,

(a+b)2=a2+2ab+b2

The distributive property is,

a(b+c)=ab+ac

The algebraic identity of cube of sum of two real numbers is,

(a+b)3=a3+3ab2+3a2b+b3

Calculation:

Consider the indefinite integral say I,

x(x2+2)2dx

Apply the algebraic identity (a+b)2=a2+2ab+b2 and then distributive property,

I=x(x2+2)2dx=x((x2)2+2(x2)(2)+22)dx=x(x4+4x2+4)dx=(x5+4x3+4x)dx

Now, apply the general power rule for integrtion in the above integral

I=(x5+15+1)+4(x3+13+1)+4x1+11+1+C1=16x6+x4+2x2

(b)

To determine

The difference between the two results in part (a).

(c)

To determine

The method which can be preferred to solve integration x(x2+2)2dx.

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