Integrating Functions over Infinite Intervals

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.