Determine whether the improper integral $\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x$ diverges or converges.
A) diverges
B) converges

Question

Determine whether the improper integral $\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x$ diverges or converges. 
A) diverges
B) converges

Quizplus · Accepted Answer

This integral is improper because the integrand goes to infinity at one of the endpoints, namely $0$. We can evaluate the integral by taking a limit: 
$$\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x = \lim _ { a \rightarrow 0 ^ { + } } \int _ { a } ^ { 1 } \frac { 3 } { x } d x = \lim _ { a \rightarrow 0 ^ { + } } \left[ 3 \ln | x | \right] _ { a } ^ { 1 } = \lim _ { a \rightarrow 0 ^ { + } } [\ln 1 - \ln a + \ln 3] = \infty. $$
Since the limit of the integral is infinite, we say that the original integral diverges.

Determine Whether the Improper Integral ∫013xdx\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x∫01​x3​dx

Determine Whether the Improper Integral $\int _ { 0 } ^ { 1 } \frac { 3 } { x } d x$