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Section 5-8 : Substitution Rule for Definite Integrals

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.