   Chapter 13.2, Problem 22E ### 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 definite integrals in Problems 1-32. ∫ 0 2 2 x 3 − 8 x 2 3   d x

To determine

To calculate: The value of the integral 022x383x2dx.

Explanation

Given Information:

The provided integral is 022x383x2dx.

Formula used:

If f is a continuous function on the closed interval [a,b], then the value of the definite integral of f that exists on the interval is,

abf(x)dx=F(b)F(a)

Where F(x)=f(x) for all x in closed interval [a,b].

The integral formula,

xndx=xn+1n+1.

Calculation:

Consider the provided integral 022x383x2dx.

Let the expression

(2x38)=u.

Differentiate with respect to x.

d(2x38)=du(6x2)dx=du

So, (6x2)dx=du

Multiply and divide the numerator by 6 and rewrite the provided integral as

022x383x2dx=02(2x38)13×66x2dx=1602(2x38)13×6x2dx

Now, substitute the values in terms of u and du and simplify the integral by the use of integral formula xndx=xn+1n+1 as,

1602(2x3<

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