Calculus 2: Improper integrals

Determine the location of the center of mass of a rod with density $\frac{1}{(x^2 + 9)^{3/2}}$ where $x$ goes from 0 to 4.

Evaluate the integral from 1 to infinity of $\displaystyle \int_{1}^{\infty} \frac{1}{x} \, dx$ and determine if it is convergent or divergent.

Integrate $\frac{1}{x^2}$ from 1 to infinity and determine if it is convergent or divergent.

Determine if the integral of $\displaystyle \int \frac{1}{(3x + 1)^2} \, dx$ is convergent or divergent.

Evaluate the improper integral $\displaystyle \int_{1}^{\infty} \frac{1}{x^2} \, dx$.

Evaluate $\lim_{{B \to \infty}} \displaystyle \int_{{1}}^{{B}} \frac{1}{x} \, dx$.

Evaluate the limit $\lim_{{B \to \infty}} \int_{{1}}^{{B}} \frac{1}{x^n} \, dx$ for any power $n$.

Evaluate the integral $\displaystyle \int_{1}^{\infty} e^{-2x} \, dx$.

Evaluate $\displaystyle \lim_{{B \to \infty}} \int_{{0}}^{{B}} \cos x \, dx$.

Evaluate $\lim_{{a \to -\infty}} \int_{{a}}^{{0}} e^x \, dx$ using integration by parts.

Evaluate the integral $\displaystyle \int_{-\infty}^{\infty} e^{x^2} \, dx$.

Evaluate the integral $\displaystyle \int_{0}^{9} \frac{1}{\sqrt{x}} \, dx$.

Evaluate the integral $\displaystyle \int_{C}^{9} x^2 \, dx$ as $C \to 0^+$.

Evaluate the integral $\displaystyle \int_{0}^{1} \ln x \, dx$ using integration by parts and limits as the variable approaches zero from the right.

Evaluate the integral $\int_{0}^{\infty} f(x) \, dx$ by breaking it into $\int_{0}^{1} f(x) \, dx$ and $\int_{1}^{\infty} f(x) \, dx$, addressing the infinite interval and discontinuity at zero.

Evaluate the integral from $1$ to $\infty$ of $\frac{1}{x} + \sin^2 x \, dx$.

Determine if the integral $\displaystyle \int_{1}^{\infty} \frac{1}{1+x} \, dx$ is convergent or divergent using the comparison test.

Determine whether the integral from negative infinity to 1 of $\frac{1}{2x-5} \, dx$ is convergent or divergent and evaluate it if it is convergent.

Determine whether the integral from 0 to infinity of $\displaystyle \int_{0}^{\infty} \frac{x}{(x^2+2)^2} \, dx$ is convergent or divergent and evaluate it if it is convergent.

Determine whether the integral from negative infinity to infinity of $\displaystyle \int_{-\infty}^{\infty} (2 - v^4) \, dv$ is convergent or divergent and evaluate if possible.