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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 5.2, Problem 42E
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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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Integration by Substitution In Exercises 35–42, use the method of substitution to find the indefinite integral. Check your result by differentiating. See Examples 6 and 7.

∫ x 2 + 3 x 3 + 9 x 3   d x

To determine

To calculate: The indefinite integral of ∫x2+2x3+9x3 dx and to check the result by differentiating.

Explanation

Given Information:

The provided expression is, ∫x2+2x3+9x3 dx.

Formula used:

General Power Rule:

If u is a differentiable function of x, then

∫undudxdx=∫undu=un+1n+1+c,n≠−1

The Constant Multiple Rule:

ddxcf(x)=cf′(x)

The Chain Rule:

ddxf(g(x))=f′(g(x))g′(x)

The Power Rule:

ddxxn=nxn−1

Where n is a real number.

Calculation:

Consider the expression, ∫x2+2x3+9x3 dx,

This can be written as ∫(x3+9x)−13(x2+2)dx.

Let u=x3+9x,

So,

du=(3x2+9) dx=3(x2+2)dx

Now use the general power rule to get,

13∫(u)−13du

Now the integral will be:

13⋅u−13+1−13+1+C=13⋅u2323+C=13⋅32u23+C=12u23+C

Substitute back the value of u to get,

∫x2</

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