Calculus 2

Evaluate the integral $\displaystyle \int_{1}^{4} \frac{1}{(4-x)^{2/3}} \, dx$.

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.

Determine whether the integral from 2 to 3 of $\displaystyle \int_{2}^{3} \frac{1}{\sqrt{3 - x}} \, dx$ is convergent or divergent and evaluate it if it is convergent.

Given the improper integral from 1 to infinity of $\displaystyle \int_{1}^{\infty} \frac{1}{x^p} \, dx$, determine if it is convergent or divergent for different values of $p$.

Use the comparison theorem to determine whether the integral from 0 to $\pi$ of $\frac{\sin^2 x}{\sqrt{x}} \, dx$ is convergent or divergent.

Determine the convergence or divergence of the integral from 1 to infinity of $\displaystyle \int_{1}^{\infty} \frac{2 + e^{-x}}{x} \, dx$ using the comparison theorem.

Find the indefinite integral of $\frac{1}{x^2 - 4}$ using integration by partial fractions.

Evaluate the integral $\displaystyle \int \frac{x - 4}{x^2 + 2x - 15} \, dx$ by performing partial fraction decomposition.

Integrate $\frac{1}{x - 1} + \frac{1}{x - 2} + \frac{1}{(x - 2)^2}$ using partial fraction decomposition, acknowledging repeated linear factors.

Find the antiderivative of $\frac{x^2 + 9}{(x^2 - 1)(x^2 + 4)}$ using partial fraction decomposition.

Find the surface area of the solid formed by revolving the curve $y=\sqrt{4x-x^2}$ about the x-axis, where x ranges from $0.5$ to $1.5$.

Find the surface area of the solid formed by revolving the curve $x = 3\sqrt{4 - y}$ about the y-axis, where $y$ ranges from 0 to $\frac{15}{4}$.

Calculate the surface area of a solid of revolution when rotating a given curve segment about the x-axis using the formula $S = \int_{a}^{b} 2\pi y \, \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx$ or $S = \int_{c}^{d} 2\pi y \, \sqrt{1 + \left( \frac{dx}{dy} \right)^2} \, dy$, depending on whether the curve is described as $y(x)$ or $x(y)$.

Given a function that forms a squiggly line, rotate it around the x-axis to form a three-dimensional shape. Find the surface area of this shape.