Determine whether $\int_{-\infty }^{\,\infty }4x^{3}\,dx$ converges or diverges.

Solution We begin by writing $\int_{-\infty }^{\,\infty }4x^{3}\,dx=\int_{-\infty }^{\,0}4x^{3}\,dx+\int_{0}^{\,\infty }4x^{3}\,dx$ and evaluate each improper integral on the right. $$\int_{-\infty }^{0}4x^{3}\,dx\hbox{:} \quad \lim_{a\,\rightarrow \,-\infty }\left( 4\int_{a}^{0}x^{3}\,dx\right) = \lim_{a\,\rightarrow \,-\infty }\left[ x^{4}\right] _{a}^{0}=\lim_{a\,\rightarrow -\infty }(0-a^{4})=-\infty$$

There is no need to continue. $\int_{-\infty }^{\,\infty }4x^{3}\,dx$ diverges.