Question

Try to evaluate\(\int {(1 + \ln x)} \sqrt {1 + {{(x\ln x)}^2}} dx\)with a computer algebra system. If it doesn't return an answer, make a substitution that changes the integral into one that the CAS can evaluate

Short answer

\(\int {(1 + \ln x)} \sqrt {1 + {{(x\ln x)}^2}} dx = \frac{{x\ln x}}{2}\sqrt {{{(x\ln x)}^2} + 1} + \frac{1}{2}{\sinh ^{ - 1}}(x\ln x){\rm{. }}\)

Step-by-step solution

Definition

Integrationis the process of finding the area of the region under the curve.

Using CAS

Use computer algebraic system to solve the following integral.

\(\int {(1 + \ln x)} \sqrt {1 + {{(x\ln x)}^2}} dx\)

In CAS put, intgeral\(\left( {1 + \ln x} \right)\sqrt {1 + {{\left( {x\ln x} \right)}^2}} \)& press enter.

From the output above it is observed that, the integral could not be solved by the CAS. So we need to rewrite it in transformed form using substitution to make the integral solvable using CAS.

Substitution

Substitute\(u = x\ln x\)in\(\int {(1 + \ln x)} \sqrt {1 + {{(x\ln x)}^2}} dx\)

Differentiating we have\(du = (1 + \ln x)dx\)

Now use these substitutions and rewrite the integral.

\(\int {(1 + \ln x)} \sqrt {1 + {{(x\ln x)}^2}} dx = \int {\sqrt {1 + {u^2}} } du\)

Now solve the integral\(\int {\sqrt {1 + {u^2}} } du\)using CAS.

Type the following command in Maple and then press ENTER to obtain the final result.

\(\int {\sqrt {1 + {u^2}} } du = \frac{1}{2}u\sqrt {{u^2} + 1} + \frac{1}{2}{\sinh ^{ - 1}}u\)

Using back substitution we have:

\(\int {(1 + \ln x)} \sqrt {1 + {{(x\ln x)}^2}} dx = \frac{{x\ln x}}{2}\sqrt {{{(x\ln x)}^2} + 1} + \frac{1}{2}{\sinh ^{ - 1}}(x\ln x){\rm{. }}\)