Integration & Antiderivatives (College Board AP® Calculus AB): Exam Questions

Work out the indefinite integral $\int \frac{x^2+x-12}{x^2-3x}dx$.

Work out the indefinite integral $\int {\left(x^4-x+\frac{1}{x}\right)}^{2}dx$.

The rate of change of a curve is given by $\frac{dy}{dx}=3x^2-\sin x$. Given that the curve passes through the point $\left(\pi ,1\right)$, find the equation of the curve in terms of $x$.

The function $A$ is used to model the surface area of a pond, in square meters, that is covered by a particular species of invasive pond weed. The rate of change function for $A$ is $A^{\text{'}}$, with the rate of change at time $t$ given by $A^{\text{'}}\left(t\right)=e^{-\frac{t}{2}}$. Given that 1 square meter of the pond is covered by the weed at at time $t=0$ , find an expression for $A\left(t\right)$.

Given that $h$ satisfies the equation $h\left(x\right)=\int \left(e^{3x}-3^x+1\right)dx$ and that when $x=1$ the value of $h$ is $\frac{1}{3}{e}^3-\frac{3}{\ln 3}$, find an explicit expression for $h$ in terms of $x$.

A function $f$ satisfies the equation $f^{\text{'}}\left(x\right)=\frac{x^2+x+1}{x^3+x}$, where $f^{\text{'}}$ is the derivative of $f$. Given that $f\left(1\right)=0$, find an expression for $f\left(x\right)$ in terms of $x$.

Given that for a particular curve $\frac{dy}{dx}=\frac{{\left(e^{2x}+e^{-2x}\right)}^2}{e^{2x}}$ and that the curve passes through the origin, find the equation of the curve.

Given that $f\text{'}\left(x\right)=\cos x{\csc }^{2}x$ and $f\left(\frac{\pi }{2}\right)=0$ find $f\left(x\right)$.

Work out the indefinite integral $\int \left(\frac{1}{\sqrt{4-4x^2}}+\frac{2}{1+x^2}\right)dx$.

Given that $f\text{'}\left(x\right)=5\times 3^{\left(2x+1\right)}$ and that $f\left(0\right)=7$, find $f\left(x\right)$.