   Chapter 12.2, Problem 26E ### 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. ∫ ( 2 x 4 + 3 ) 2 ( 8 x   ​ d x )

To determine

To calculate: The value of the integral (2x4+3)28xdx and also check the solution by differentiation.

Explanation

Given Information:

The provided integral is (2x4+3)28xdx.

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,

(2x4+3)28xdx

Now, let 2x4+3=u

Differentiate 2x4+3=u with respect to x and get,

8x3dx=du

Here, this substitution does not work.

Hence, this method fails.

So, simplify the integrand algebraically as,

(2x4+3)28xdx=(4x8+12x4+9)8xdx=(32x9+96x5+72x)dx

Use the power rule of integrals:

undu=un+1n+1+C

Now, integrate the above expression with respect to x as,

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