Evaluate the integral .

Question

Evaluate the integral  .

Quizplus · Accepted Answer

The integral is of the form $\int_{0}^{1}\frac{x^{a-1}}{1-x^{c}}dx$, where $a$ and $c$ are positive constants. This integral diverges if $a\geq c$, and converges if $a<c$. In this case, $a=2$ and $c=3$, so $a<c$ and the integral converges. However, the integrand has a vertical asymptote at $x=1$, which means the integral is improper. We can use the substitution $u=1-x^3$, $du=-3x^2 dx$ to obtain $$\int_{0}^{1}\frac{dx}{(1-x)(1+x+x^2)}=\int_{0}^{1}\frac{-du}{3u}=-\frac{1}{3}\ln|u|\bigg|_{0}^{1}.$$ Since $\ln|0|$ is undefined, the integral diverges. Therefore, the answer is (E) integral diverges.

Evaluate the Integral