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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 an Improper Integral In Exercises 7-20, determine whether the improper integral diverges or converges. Evaluate the integral if it converges. See Examples 1, 2, and 3.

4 1 x ( I n x ) 3 d x

To determine

Whether the improper integral 41x(lnx)3dx diverges or converges and evaluate if it converges.

Explanation

Given Information:

The provided expression is, 41x(lnx)3dx.

From definition of improper integral.

af(x)dx=limbabf(x)dx

Also, the expression for the integration of a inverse function is as follows:

1xdx=|x|+C

Consider the provided expression:

41x(lnx)3dx

The improper integral converges if the limit exists otherwise the improper integral diverges.

Use the property of improper integral and simplify as:

41x(lnx)3dx=limb4b1x(lnx)3dx

Integrate the integrand by substation method as:

Assume lnx=u

Differentiate as:

1xdx=du

Now, substitute the values and integrate as:

1x(lnx)3dx=1u3du=12u2+C

Again, substitute the value of u

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