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

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Chapter 5.2, Problem 38E
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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.

∫ t t 2 + 1   d t

To determine

To calculate: The indefinite integral of ∫tt2+1 dt and to check the result by differentiating.

Explanation

Given Information:

The provided expression is, ∫tt2+1 dt.

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, ∫tt2+1 dt.

Let u=t2+1,

So,

du=2t dt12⋅dudt=t

Now use the general power rule to get,

12∫(u)12 du

Now the integral will be:

12⋅u12+112+1+C=12⋅u3232+C=12⋅23u32+C=13u32+C

Substitute back the value of u

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