   Chapter 13.6, Problem 15E ### 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. ∫ 4 0 x 3 x 2 + 9   d x

To determine

To calculate: The value of the integral 04x3x2+9dx.

Explanation

Given Information:

The provided integral is 04x3x2+9dx.

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,

04x3x2+9dx

Multiply and divide by 2 and rewrite the integral as,

1204x2x2+92xdx

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=(x2+9)1/22xdx

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=(x2+9)1/22xdx

Now, Let x2+9=t,

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

2xdx=dt

Thus, the integral becomes,

v=(x2+

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