Evaluate $\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }$ .
A)1
B) $\frac { 1 } { 3 }$
C)Diverges
D)0

Question

Quizplus · Accepted Answer

This integral tests the convergence of an improper integral of the form $\int _ { a } ^ { \infty } f(x) dx$. To evaluate it, we rewrite the integral as $\int _ { 1 } ^ { \infty } x^{-1/3} dx$. The antiderivative of $x^{-1/3}$ is $\frac{3}{2}x^{2/3}$, and we evaluate this from 1 to infinity. As the upper limit approaches infinity, the value of $\frac{3}{2}x^{2/3}$ also approaches infinity, indicating that the integral diverges.

Evaluate ∫1∞dxx3\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }∫1∞​3x​dx​

Evaluate $\int _ { 1 } ^ { \infty } \frac { d x } { \sqrt [ 3 ] { x } }$