Calculus I - Substitution Rule for Definite Integrals

Show General Notice Show Mobile Notice Show All Notes Hide All Notes

General Notice

I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.

Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.

Paul
February 18, 2026

Mobile Notice

You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 5.8 : Substitution Rule for Definite Integrals

Back to Problem List

8. Evaluate the following integral, if possible. If it is not possible clearly explain why it is not possible to evaluate the integral.

\[ \int_{1}^{5}{{\frac{{2{x^3} + x}}{{{x^4} + {x^2} + 1}} - \frac{x}{{{x^2} - 4}}\,dx}}\] Show Solution

Be very careful with this problem. Recall that we can only do definite integrals if the integrand (i.e. the function we are integrating) is continuous on the interval over which we are integrating.

In this case the second term has division by zero at \(x = 2\) and so is not continuous on \(\left[ {1,5} \right]\) and therefore this integral can’t be done.