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

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Chapter 5.2, Problem 34E
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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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Applying the General Power Rule In Exercises 9–34, find the indefinite integral. Check your result by differentiating. See Examples 1, 2, 3, and 5.

∫ 3 x 2 1 − x 3 d x

To determine

To calculate: The indefinite integral of ∫3x21−x3 dx and to check the result by differentiating.

Explanation

Given Information:

The provided expression is ∫3x21−x3 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 ∫3x21−x3 dx,

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

Let u=1−x3,

So,

dudx=−3u2

Now use the general power rule to get,

∫−(u)−12du

Now the integral will be:

−u−12+1−12+1+C=−u1212+C=−2

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