Methods of Integration (College Board AP® Calculus BC): Exam Questions

Find the value of ${\int }_0^{4}x\sqrt{16-x^2}dx$.

Show that ${\int }_1^{\sqrt{3}}\frac{1}{\left(1+x^2\right)\arctan x}dx=\ln \left(\frac{a}{b}\right)$, where $a$ and $b$ are integers to be found.

Solve the indefinite integral $\int \frac{12}{\sqrt{12-4x-x^2}}dx$.

Let the function $f$ be defined by $f\left(x\right)=x^2{\sec }^2\left(x^3\right)$.

Let $g$ be a function whose derivative is given by $g^{\text{'}}\left(x\right)=f\left(x\right)$ on the interval $\left[0,1\right]$.

Given that $g\left(0\right)=\pi$, find $g\left(\frac{1}{2}\right)$.

Find the value of ${\int }_{-1}^1\frac{2x+3}{x^2+2x+2}dx$. Give the answer in the form $\ln \left(a\right)+\arctan \left(b\right)$, where $a$ and $b$ are constants to be found.

Let $h$ be the function given by $h\left(x\right)=\frac{1}{x^{2/3}}$.

In general, the average value of a function $f$ on the unbounded interval $\left[a,\infty \right)$ is defined to be $\underset{b\to \infty }{\lim }\left[\frac{{\int }_a^{b}f\left(x\right)\mathit{d}x}{b-a}\right]$.

Show that the improper integral ${\int }_5^{\infty }h\left(x\right)\mathit{d}x$ is divergent, but the average value of $h$ on the interval $\left[5,\infty \right)$ is finite.

Find the value of ${\int }_0^6\frac{3x+5}{x^2+3x+2}\mathit{d}x$ in the form $\ln a$ where $a$ is an integer.

Given that $f\text{'}\left(x\right)=4x^3\ln x$ and that the graph $y=f\left(x\right)$ passes through $\left(1,0\right)$, find $f\left(x\right)$.

Find $\int (t+5)\text{cos}\left(\frac{t}{4}\right)\text{d}t$. Show the work that leads to your answer.

Water is pumped into a swimming pool at a constant rate of 10 gallons per minute. Water leaks out of the pool at the rate of $\frac{3t+13}{t+5}$ gallons per minute, for $0\le t\le 180$ minutes. At the time $t=0$, the swimming pool contains 15 gallons of water.

How many gallons of water leak out of the swimming pool from time $t=0$ to $t=5$ minutes?

How many gallons of water are in the swimming pool at time $t=5$ minutes?

Write an expression for $V\left(t\right)$, the total number of gallons of water in the swimming pool at time $t$. Write the function in the form $a+bt+k\ln \left(f\left(t\right)\right)$, where $a$, $b$ and $k$ are constants to be found and $f$ is a function of $t$.

Let $f$ be a function defined by $f\left(x\right)=\{\begin{array}{ll}\frac{x}{4x^2+1}& x<0\\ x{e}^{x^2+1}& x\ge 0\end{array}$

Find the exact value of ${\int }_{-1}^{2}f\left(x\right)\mathit{d}x$.

Solve the indefinite integral $\int \frac{2x^3-x^2+10x-1}{x^2+4}\mathit{d}x$.

Let $g$ be the piecewise-linear function defined on $\left[-4,6\right]$ whose graph is given above. Let $f\left(x\right)=g\left(x\right)+\sin \left(\frac{\pi x}{4}\right)e^{\cos \left(\frac{\pi x}{4}\right)}$.

Find ${\int }_{-4}^{6}f\left(x\right)dx$.

Solve the indefinite integral $\int \frac{e^x}{\sqrt{8+2e^x-e^{2x}}}\mathit{d}x$.

Evaluate the improper integral ${\int }_5^{\infty }\frac{4}{3x^2-10x+7}\mathit{d}x$, or show that the integral diverges.

Find $\int \ln xdx$. Show all of the work that leads to your answer.

Find $\int \arctan xdx$. Show all of the work that leads to your answer.

Find $\int e^x\sin xdx$. Show all the work that leads to your answer.