Tutorial Exercise

Evaluate the definite integral. Use a graphing utility to verify your result.

${\displaystyle \underset{8}{\overset{9}{}}}(\frac{4}{x^2}-1)dx$

Step $1$

To evaluate the integral ${\displaystyle \underset{8}{\overset{9}{}}}(\frac{4}{x^2}-1)dx,$ first use the property of definite integrals, that is$,$ the following.

${\displaystyle \underset{a}{\overset{b}{}}}(f(x)+g(x))dx={\displaystyle \underset{a}{\overset{b}{}}}f(x)dx+{\displaystyle \underset{a}{\overset{b}{}}}g(x)dx$

Therefore, we get the following.

${\displaystyle \underset{8}{\overset{9}{}}}(\frac{4}{x^2}-1)dx={\displaystyle \underset{8}{\overset{9}{}}}\frac{x^2}{9}dx-{\displaystyle \underset{8}{\overset{9}{}}}(1)dx$