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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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.

( 3   -   x ) 2   d x

(a)

To determine

To calculate: The value of the provided indefinite integral (3x)2dx using simple power rule and general power rule.

Explanation

Given Information:

The provided indefinite integral is (3x)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 difference of two real numbers is,

(ab)2=a22ab+b2

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

(ab)3=a3+3ab23a2bb3

Calculation:

Consider the indefinite integral say I,

(3x)2dx

Apply the algebraic identity (ab)2=a22ab+b2 and then distributive property,

I=(3x)2dx=(9+x26x)dx

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

I=9x+(x2+12+1)6x1+11+1

(b)

To determine

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

(c)

To determine

The method which can be preferred to solve integration (3x)2dx.

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