   Chapter 12.2, Problem 28E ### 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

# Evaluate the integrals in Problems 7-36. Check your results by differentiation. ∫ x 2 + 2 x ​ 3  ( x + 1 ) d x

To determine

To calculate: The value of the integral x2+2x3(x+1)dx and also check the solution by differentiation.

Explanation

Given Information:

The provided integral is x2+2x3(x+1)dx.

Formula used:

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

undu=un+1n+1+C

According to the power rule of derivative,

ddx(xn)=nxn1

Calculation:

Consider the provided integral,

x2+2x3(x+1)dx

Rewrite the integral by multiplying and dividing by 2 as,

12x2+2x3(2x+2)dx

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.

Let, u=x2+2x and n=13

Differentiate u=x2+2x with respect to x and get,

du=(2x+2)dx

Now, all required parts are present, so the integral is of the form,

12x2+2x3(2x+2)dx=12u1/3du=12

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