   Chapter 13.6, Problem 14E ### 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

# In Problems 1-16, use integration by parts to evaluate the integral. ∫ x 3 9 − x 2   d x

To determine

To calculate: The value of the integral x39x2dx.

Explanation

Given Information:

The provided integral is x39x2dx.

Formula used:

The formula for integration by parts is given by:

udv=uvvdu

According to the power rule of integrals,

xndx=xn+1n+1+C

Differentiation formula,

d(xn)dx=nxn1

Calculation:

Consider the provided integral,

x39x2dx

Rewrite the integral as,

x2x9x2dx

Now, split the integrand into two parts

Set one part equal to u and another part equal to dv.

So, the values will be:

u=x2

And,

dv=x9x2dx

Now use the differential formula d(xn)dx=nxn1 to differentiate u=x2

du=2xdx

And integrate v by the use of power rule of integrals

v=x9x2dx

Now, let 9x2=t,

And differentiate the equation 9x2=t by the use of differential formula d(xn)dx=nxn1

2xdx=dtxdx=dt2

Thus, the integral becomes,

v=x9x2d<

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