   Chapter 5.2, Problem 37E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### 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 ( 6 x 2 − 7 ) 3 d x

To determine

To calculate: The indefinite integral of x(6x27)3dx and to check the result by differentiating.

Explanation

Given Information:

The provided expression is, x(6x27)3dx.

Formula used:

General Power Rule:

If u is a differentiable function of x, then

undudxdx=undu=un+1n+1+c,n1

The Constant Multiple Rule:

ddxcf(x)=cf(x)

The Chain Rule:

ddxf(g(x))=f(g(x))g(x)

The Power Rule:

ddxxn=nxn1

Where n is a real number.

Calculation:

Consider the expression, x(6x27)3dx,

Let u=6x27,

So,

du=12xdx112dudx=x

Now use the general power rule to get,

112(u)3du

Now the integral will be:

112u3+13+1+C=112u44+C=148u4+C

Substitute back the value of u to get,

x(6x2

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