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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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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.

1 e 2 x ln x d x

To determine

To calculate: The value of definite integral 1e2xlnxdx.

Explanation

Given Information:

The provided definite integral is 1e2xlnxdx.

Formula used:

The integration by parts udv=uvvdu.

Where, u and v are function of x.

xndx=xn+1n+1+C,n1

Calculation:

Consider the definite integral 1e2xlnxdx

Here,

dv=2xdx and u=lnx

First find v,

dv=2xdxdv=2xdx

On further solving,

v=x2 …...…... (1)

Find du:

u=lnx

Differentiate both side with respect x;

dudx=d(lnx)dxdudx=1x

And,

du=1xdx …...…... (2)

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

2xlnxdx=[x2lnx]x21xdx

Now, take the lower limit as 1 and upper limit as e,

1

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