AP Calculus BC Unit 6 (BC topics): Integration by Parts, Partial Fractions and Improper Integrals

The three integration ideas BC adds to Unit 6: integration by parts for products, partial fractions for rational functions with distinct linear factors, and improper integrals, which are limits of definite integrals over unbounded intervals or across vertical asymptotes.

This page covers only the BC topics of Unit 6. The rest of the unit is the same as AP Calculus AB: AB Unit 6 notes and practice.

BC topics on this page: 6.11 Integration by parts; 6.12 Linear partial fractions; 6.13 Improper integrals.

Key formulas and facts

• Integration by parts: $\int u\,dv=uv-\int v\,du$; for definite integrals, $\int_a^bu\,dv=\big[uv\big]_a^b-\int_a^bv\,du$
• Choosing $u$: pick the factor that gets simpler when differentiated (a power of $x$, $\ln x$, $\arctan x$); $dv$ is the rest, and must be something you can integrate
• Distinct linear factors: $\frac{px+q}{(x-r)(x-s)}=\frac{A}{x-r}+\frac{B}{x-s}$, and $\int\frac{A}{x-r}\,dx=A\ln|x-r|+C$
• If the numerator's degree is at least the denominator's, divide first
• Improper integrals: $\int_a^{\infty}f(x)\,dx=\lim_{b\to\infty}\int_a^bf(x)\,dx$; if $f$ is unbounded at $c$ in $[a,b]$, split at $c$ and take one-sided limits
• $p$-integrals: $\int_1^{\infty}\frac{dx}{x^p}$ converges exactly when $p>1$; $\int_0^1\frac{dx}{x^p}$ converges exactly when $p<1$

Worked examples

• Parts: $\int x\ln x\,dx$ with $u=\ln x$, $dv=x\,dx$ gives $\frac{x^2}{2}\ln x-\frac{x^2}{4}+C$
• Parts with $dv=dx$: $\int\ln x\,dx=x\ln x-x+C$
• Partial fractions: $\frac{7x-1}{(x-1)(x+2)}=\frac{2}{x-1}+\frac{5}{x+2}$ (cover up $x=1$: $\frac{6}{3}=2$; cover up $x=-2$: $\frac{-15}{-3}=5$)
• Improper, convergent: $\int_0^{\infty}e^{-3x}\,dx=\lim_{b\to\infty}\frac{1-e^{-3b}}{3}=\frac13$
• Improper, divergent: $\int_1^{\infty}\frac{dx}{x}=\lim_{b\to\infty}\ln b=\infty$
• Asymptote at an endpoint: $\int_0^4\frac{dx}{\sqrt x}=\lim_{a\to0^+}\left(4-2\sqrt a\right)=4$

Common traps

• Choosing $u=e^x$ in $\int xe^x\,dx$: the new integral $\int\frac{x^2}{2}e^x\,dx$ is harder, not easier.
• Dropping the minus sign in $uv-\int v\,du$, especially on the second round of a repeated integration by parts.
• Integrating across a vertical asymptote as if it were not there: $\int_{-1}^1\frac{dx}{x^2}$ is not $-2$; it diverges.
• Writing $\infty$ as a number in the antiderivative: always write the limit as $b\to\infty$ and evaluate it.
• Forgetting to divide before using partial fractions when the numerator's degree is not less than the denominator's.
• Saying an improper integral 'equals infinity, so it converges': an infinite limit means the integral diverges.

Practice questions

22 multiple-choice questions and 2 free-response questions. Work each one out before you open the answer. In a free-response scoring guide, each line is worth the points it states. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

Which of the following is $\displaystyle\int x e^{3x}\,dx$?

1. $\left(\frac{x}{3}+\frac19\right)e^{3x}+C$
2. $\frac{x^2}{2}\cdot\frac{e^{3x}}{3}+C$
3. $\left(\frac{x}{3}-\frac19\right)e^{3x}+C$
4. $(3x-1)e^{3x}+C$

Question 2 · Multiple choice

Evaluate $\displaystyle\int_1^{e} x^4\ln x\,dx$ exactly.

1. $\frac{e^{5}}{5}$
2. $\frac{1}{25} + \frac{4 e^{5}}{25}$
3. $- \frac{1}{25} + \frac{4 e^{5}}{25}$
4. $\frac{4}{25} + \frac{e^{5}}{25}$

Question 3 · Multiple choice

Which of the following is $\displaystyle\int x\cos(2x)\,dx$?

1. $\frac{x}{2}\sin(2x)-\frac14\cos(2x)+C$
2. $2x\sin(2x)+\cos(2x)+C$
3. $\frac{x^2}{4}\sin(2x)+C$
4. $\frac{x}{2}\sin(2x)+\frac14\cos(2x)+C$

Question 4 · Multiple choice

What is the value of $\displaystyle\int_0^1 \arctan x\,dx$?

1. $\frac{\pi}{4}$
2. $\frac{\ln{\left(2 \right)}}{2} + \frac{\pi}{4}$
3. $- \frac{\ln{\left(2 \right)}}{2} + \frac{\pi}{4}$
4. $- \frac{1}{2} + \frac{\pi}{4}$

Question 5 · Multiple choice

Which of the following is $\displaystyle\int x^2e^{-x}\,dx$?

1. $-(x^2+2x+2)e^{-x}+C$
2. $-(x^2-2x+2)e^{-x}+C$
3. $(x^2+2x+2)e^{-x}+C$
4. $-\frac{x^3}{3}e^{-x}+C$

Question 6 · Multiple choice

The function $f$ has a continuous second derivative. The table gives values of $f$ and $f'$. What is the value of $\displaystyle\int_1^4 x f''(x)\,dx$?

1. $17$
2. $22$
3. $2$
4. $12$

Question 7 · Multiple choice

An antiderivative of $e^{2x}\cos x$ is needed. Which of the following is $\displaystyle\int e^{2x}\cos x\,dx$?

1. $\frac15e^{2x}(2\cos x-\sin x)+C$
2. $\frac15e^{2x}(2\cos x+\sin x)+C$
3. $\frac15e^{2x}(\cos x+2\sin x)+C$
4. $\frac13e^{2x}(2\cos x+\sin x)+C$

Question 8 · Multiple choice

Which of the following is $\displaystyle\int\frac{5}{(x-2)(x+3)}\,dx$?

1. $\ln|x+3|-\ln|x-2|+C$
2. $5\ln|x-2|-5\ln|x+3|+C$
3. $5\ln|(x-2)(x+3)|+C$
4. $\ln|x-2|-\ln|x+3|+C$

Question 9 · Multiple choice

What is the value of $\displaystyle\int_0^1\frac{x+7}{(x+1)(x+3)}\,dx$?

1. $\ln{\left(6 \right)}$
2. $\ln{\left(\frac{9}{2} \right)}$
3. $\ln{\left(\frac{3}{2} \right)}$
4. $\ln{\left(\frac{27}{8} \right)}$

Question 10 · Multiple choice

The constants $A$ and $B$ satisfy $\dfrac{3x+11}{x^2-x-6}=\dfrac{A}{x-3}+\dfrac{B}{x+2}$ for all $x$ other than $3$ and $-2$. What is $A$?

1. $4$
2. $-1$
3. $3$
4. $\frac{11}{5}$

Question 11 · Multiple choice

Which of the following is $\displaystyle\int\frac{x^2}{x^2-1}\,dx$?

1. $\frac12\ln|x-1|-\frac12\ln|x+1|+C$
2. $x-\frac12\ln|x-1|+\frac12\ln|x+1|+C$
3. $x+\frac12\ln|x-1|-\frac12\ln|x+1|+C$
4. $\frac{x^3}{3}\ln|x^2-1|+C$

Question 12 · Multiple choice

What is the value of $\displaystyle\int_2^{\infty}\frac{1}{x(x-1)}\,dx$?

1. $\frac{1}{2}$
2. $\ln{\left(2 \right)}$
3. $\ln{\left(\frac{3}{2} \right)}$
4. The integral diverges.

Question 13 · Multiple choice

Which of the following is $\displaystyle\int\frac{1}{x(10-x)}\,dx$ for $0<x<10$?

1. $\frac{1}{10}\ln\big(x(10-x)\big)+C$
2. $10\ln\left(\frac{x}{10-x}\right)+C$
3. $\ln\left(\frac{10-x}{x}\right)+C$
4. $\frac{1}{10}\ln\left(\frac{x}{10-x}\right)+C$

Question 14 · Multiple choice

What is the value of $\displaystyle\int_3^5\frac{4}{x^2-4}\,dx$?

1. $\ln{\left(\frac{7}{15} \right)}$
2. $2 \ln{\left(\frac{15}{7} \right)}$
3. $\ln{\left(\frac{15}{7} \right)}$
4. $\ln{\left(\frac{21}{5} \right)}$

Question 15 · Multiple choice

What is the value of $\displaystyle\int_1^{\infty}\frac{6}{x^3}\,dx$?

1. $3$
2. $6$
3. $2$
4. The integral diverges.

Question 16 · Multiple choice

What is the value of $\displaystyle\int_0^{\infty}x e^{-2x}\,dx$?

1. $\frac{1}{4}$
2. $\frac{1}{2}$
3. $1$
4. The integral diverges.

Question 17 · Multiple choice

What is the value of $\displaystyle\int_0^{8}x^{-1/3}\,dx$?

1. $3$
2. $12$
3. $6$
4. The integral diverges.

Question 18 · Multiple choice

Which of the following improper integrals converge? I. $\displaystyle\int_1^{\infty}\frac{1}{\sqrt{x}}\,dx$ II. $\displaystyle\int_1^{\infty}\frac{1}{x\sqrt{x}}\,dx$ III. $\displaystyle\int_0^{1}\frac{1}{x^2}\,dx$

1. I only
2. II only
3. I and II only
4. II and III only

Question 19 · Multiple choice

What is $\displaystyle\int_{-1}^{1}\frac{1}{x^2}\,dx$?

1. $-2$
2. $0$
3. $2$
4. The integral diverges.

Question 20 · Multiple choice

For which values of $p$ does $\displaystyle\int_1^{\infty}\frac{1}{x^{p}}\,dx$ converge?

1. $p\ge1$ only
2. $p<1$ only
3. $p>1$ only
4. $0<p<1$ only

Question 21 · Multiple choice

What is the value of $\displaystyle\int_0^{\infty}\frac{1}{x^2+4}\,dx$?

1. $\frac{\pi}{4}$
2. $\frac{\pi}{2}$
3. $\frac{\pi}{8}$
4. The integral diverges.

Question 22 · Multiple choice

The graph of $y=\dfrac{3}{\sqrt{x}\,(1+x)}$ has a vertical asymptote at $x=0$. The region between the graph and the $x$-axis for $0<x\le9$ is unbounded. Using a graphing calculator where needed, what is the area of this region, or is it infinite?

1. $2.782$
2. $9.425$
3. The area is infinite.
4. $7.494$

Question 23 · Free response (9 points)

Let $g(x)=\dfrac{x+5}{x^2+x-2}$ for $x>1$.

1. (a) Find constants $A$ and $B$ so that $g(x)=\dfrac{A}{x-1}+\dfrac{B}{x+2}$. [2 points]
2. (b) Find the exact value of $\displaystyle\int_2^5 g(x)\,dx$. [2 points]
3. (c) Determine whether $\displaystyle\int_2^{\infty} g(x)\,dx$ converges or diverges. Show the work that leads to your answer. [2 points]
4. (d) Find the exact value of $\displaystyle\int_2^3\ln(x-1)\,dx$. [3 points]

Question 24 · Free response (9 points)

A medicine enters a patient's bloodstream at a rate of $r(t)=8te^{-t}$ milligrams per hour, where $t$ is the time in hours after the dose ($t\ge0$).

1. (a) Find the antiderivative $F$ of $r$ with $F(0)=0$. [2 points]
2. (b) How many milligrams enter the bloodstream during the first $2$ hours? Give an exact answer. [2 points]
3. (c) Find the total amount of medicine that enters the bloodstream over all time, $\displaystyle\int_0^{\infty}r(t)\,dt$, or show that it is infinite. [3 points]
4. (d) At what time is the rate $r(t)$ greatest, and what is that greatest rate? Justify your answer. [2 points]