   Chapter 12.2, Problem 40E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Each of Problems 41 and 42 has the form ∫ f ( x ) dx.(a) Evaluate each integral to obtain a family of functions.(b) Find and graph the family member that passes through the point (0,2). Call that function F ( x ) .(c) Find any x-values where f ( x ) is not defined but F ( x ) is.(d) At the x-values found in part (c), what kind of tangent line does F ( x ) have? ∫ x 2 d x ( x 3 − 1 ) 1 / 3

(a)

To determine

To calculate: The value of integral x2dx(x31)1/3.

Explanation

Given Information:

The provided integral is,

x2dx(x31)1/3

Formula used:

According to the power formula of integrals, if u=u(x), then:

undu=un+1n+1+C

Calculation:

Consider the provided integral:

x2dx(x31)1/3

Rewrite the provided integral by multiplying and dividing by 3 as,

133x2dx(x31)1/3

Consider the power rule of integrals:

undu=un+1n+1+C

Now, to use the power rule, the integrand should have the function u(x) and its derivative u(x) and n1.

Here,

u=x31

Then, on obtaining differentials,

du=3x2

Thus,

u=x31n=13u=3x2

All required parts are present, so the integral is of the form,

133x2dx(x31)1/3

(b)

To determine

To graph: The family member of the solution to part (a) which passes through point (0,2).

(c)

To determine

The values where the function f(x) is not defined but F(x) is if f(x)=x2(x31)1/3 and F(x)=12(x31)2/3+32.

(d)

To determine

The nature of tangent line of F(x) at the point obtained in part (c) which is x=1.

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