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

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Chapter 5.2, Problem 41E
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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 + 1 x 3 + 3 x + 4   d x

To determine

To calculate: The indefinite integral of ∫x2+1x3+3x+4 dx and to check the result by differentiating.

Explanation

Given Information:

The provided expression is, ∫x2+1x3+3x+4 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+1x3+3x+4 dx,

This can be written as ∫(x3+3x+4)−12(x2+1)dx.

Let u=x3+3x+4,

So,

du=(3x2+3) dx=3(x2+1)dx

Now use the general power rule to get,

13∫(u)−12du

Now the integral will be:

13⋅u−12+1−12+1+C=13⋅u1212+C=23⋅u12+C

Substitute back the value of u to get,

∫x2+1

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