Determine whether the improper integral converges or diverges.
-$$\int _ { - \infty } ^ { \infty } \frac { d x } { \sqrt { 5 x ^ { 6 } + 1 } }$$
A) Diverges
B) Converges

Question

Quizplus · Accepted Answer

We can use the comparison test with $\frac{1}{|x^3|}$ to show convergence:
\begin{align*}
\lim\limits_{tightarrow\infty}\int_{-t}^t \frac{dx}{\sqrt{5x^6+1}} &= \lim\limits_{tightarrow\infty}\int_{-t}^t \frac{1}{\sqrt{5}\cdot |x^3|\sqrt{1+\frac{1}{5x^6}}} \
&\leq \lim\limits_{tightarrow\infty}\int_{-t}^t \frac{1}{\sqrt{5}\cdot |x^3|} \
&=\lim\limits_{tightarrow\infty}\int_{-t}^0 \frac{1}{\sqrt{5}\cdot (-x)^{3/2}} + \int_{0}^t \frac{1}{\sqrt{5}\cdot x^{3/2}}\
&= \frac{2}{\sqrt{5}}\lim\limits_{tightarrow\infty}\left[\sqrt{x}ight]_0^t\
&= \frac{4}{\sqrt{5}}\cdot\infty
\end{align*}
Thus, since $\int_{-\infty}^\infty \frac{dx}{|x^3|}$ diverges, by the comparison test, the original integral also diverges.

Determine Whether the Improper Integral Converges or Diverges ∫−∞∞dx5x6+1\int _ { - \infty } ^ { \infty } \frac { d x } { \sqrt { 5 x ^ { 6 } + 1 } }∫−∞∞​5x6+1​dx​

Determine Whether the Improper Integral Converges or Diverges $\int _ { - \infty } ^ { \infty } \frac { d x } { \sqrt { 5 x ^ { 6 } + 1 } }$