   Chapter 6.1, Problem 46E ### 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 2 x 2 e 3 x d x

To determine

To calculate: The value of definite integral 12x2e3xdx.

Explanation

Given Information:

The provided definite integral is 12x2e3xdx.

Formula used:

The integration by parts udv=uvvdu.

Where, u and v are function of x.

eaxdx=eaxa+C

Calculation:

Consider the definite integral 12x2e3xdx

The above definite integral can be written as,

12x2e3xdx=12x2e3xdx

Here,

dv=e3xdx and u=x2

First find v,

dv=e3xdxdv=e3xdx

On further solving,

v=13e3x …...…... (1)

Now find du,

u=x2

Differentiate both side with respect x,

dudx=d(x2)dxdudx=2x

And,

du=2xdx …...…... (2)

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

x2e3xdx=13x2e3x+23xe3xdx

Use integration by parts on the second integral xe3xdx

Here,

dv=e3xdx and u=x

First find v,

dv=e3xdxdv=e3xdx

On further solving,

v=13e3x

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