Provide an Appropriate Response $\int _ { 0 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$

Essay

Provide an appropriate response.
-(a) Show that $\int _ { 0 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$ converges.
(b) Show that $\int _ { 50 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 } \leq 0.0002$ .
(c) Suppose $\int _ { 0 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$ is approximated by $\int _ { 0 } ^ { 50 } \frac { d x } { x ^ { 3 } + 1 }$ . Based on your answer to part (b), what is the maximum possible error?
(d) Use a numerical method to estimate the value of $\int _ { 0 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$ .
(e) Determine whether $\int _ { - 1 } ^ { \infty } \frac { d x } { x ^ { 3 } + 1 }$ converges or diverges, and justify your answer. If it converges, estimate its value to an accuracy of at least two decimal places.