   Chapter 6.1, Problem 43E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Evaluating a Definite Integral In Exercises 39-46, use integration by parts to evaluate the definite integral. See Example 5. ∫ 0 8 x x + 1   d x

To determine

To calculate: The value of definite integral 08xx+1dx.

Explanation

Given Information:

The provided definite integral is 08xx+1dx.

Formula used:

The integration by parts udv=uvvdu.

Where, u and v are function of x.

(ax+b)ndx=1a(ax+b)n+1n+1,n1

Calculation:

Consider the definite integral 08xx+1dx

The above definite integral can be written as,

08xx+1dx=08x(x+1)12dx

Here,

dv=(x+1)12dx and u=x

First find v,

dv=(x+1)12dxdv=(x+1)12dx

On further solving,

v=23(x+1)32 …...…... (1)

Find du:

u=x

Differentiate both side with respect x;

dudx=d(x)dxdudx=1

And,

du=1dx …...…... (2)

Apply integration by parts formula and substitute equation (1) and (2) in udv=uvvdu,

xx+1dx=23x(x+1)3223(x+1)32dx

Now, take the lower limit as 0 and upper limit as 8,

08xx+1dx=[23x(x+1)32

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