Answer
\[ = 0\]
Work Step by Step
\[\begin{gathered}
\int_{ - \pi /4}^{\pi /4} {{{\tan }^3}x{{\sec }^2}xdx} \hfill \\
\hfill \\
use\,\,{\text{ }}\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C} \hfill \\
\hfill \\
= \,\,\left[ {\frac{{{{\tan }^4}x}}{4}} \right]_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} \hfill \\
\hfill \\
evaluate\,\,the\,limits \hfill \\
\hfill \\
= \frac{{{{\tan }^4}\,\left( {\frac{\pi }{4}} \right)}}{4} - \frac{{{{\tan }^4}\,\,\,\left( { - \frac{\pi }{4}} \right)}}{4} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{{\,{{\left( 1 \right)}^4}}}{4} - \frac{{\,{{\left( { - 1} \right)}^4}}}{4} \hfill \\
\hfill \\
\hfill \\
= \frac{1}{4} - \frac{1}{4} = 0 \hfill \\
\end{gathered} \]
