AP Calculus AB Unit 6: Integration and Accumulation of Change

Accumulation as the area under a rate, left, right, midpoint and trapezoidal sums (including from tables with unequal widths), definite integral notation and limits of Riemann sums, the Fundamental Theorem of Calculus in both forms, accumulation functions, properties of definite integrals, antiderivatives, substitution, and long division or completing the square before integrating.

CED topics in this unit: 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 6.10, 6.14.

Key formulas and facts

• $\int_a^br(t)\,dt$ is the net change in the quantity whose rate is $r$
• Left sum $\sum_{i=0}^{n-1}f(x_i)\Delta x_i$; right sum $\sum_{i=1}^{n}f(x_i)\Delta x_i$; trapezoid $\sum\frac{f(x_{i-1})+f(x_i)}{2}\Delta x_i$
• $\int_a^bf(x)\,dx=\lim_{n\to\infty}\sum_{i=1}^{n}f\left(a+\frac{(b-a)i}{n}\right)\frac{b-a}{n}$
• FTC (first form): $\frac{d}{dx}\int_a^{g(x)}f(t)\,dt=f(g(x))\,g'(x)$
• FTC (second form): $\int_a^bf(x)\,dx=F(b)-F(a)$ where $F'=f$
• $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ ($n\ne-1$), $\int\frac1x\,dx=\ln|x|+C$, $\int\frac{1}{1+x^2}\,dx=\arctan x+C$
• $\int_a^bf=-\int_b^af$ and $\int_a^bf+\int_b^cf=\int_a^cf$

Worked examples

• Trapezoids from a table: $t=0,2,6$ with $r=4,7,3$ gives $2\cdot\frac{4+7}{2}+4\cdot\frac{7+3}{2}=11+20=31$
• Limit of a sum: $\lim_{n\to\infty}\sum_{i=1}^n\left(1+\frac{2i}{n}\right)^2\frac2n=\int_1^3x^2\,dx=\frac{26}{3}$
• FTC with the chain rule: $\frac{d}{dx}\int_0^{x^3}\cos t\,dt=3x^2\cos(x^3)$
• Substitution: $\int_0^1x(x^2+1)^3\,dx$ with $u=x^2+1$ is $\frac12\int_1^2u^3\,du=\frac{15}{8}$
• Completing the square: $\int\frac{dx}{x^2+4x+5}=\int\frac{dx}{(x+2)^2+1}=\arctan(x+2)+C$
• Long division: $\int\frac{x^2}{x+1}\,dx=\int\left(x-1+\frac{1}{x+1}\right)dx=\frac{x^2}{2}-x+\ln|x+1|+C$

Common traps

• Using equal widths in a Riemann sum when the table's intervals are unequal.
• Forgetting to change the limits (or to substitute back) after a $u$-substitution.
• Dropping the factor $g'(x)$ when differentiating $\int_a^{g(x)}f(t)\,dt$.
• Treating a definite integral as total area when the function goes below the axis.
• Forgetting $+C$ on an indefinite integral.
• Over or under: a left sum underestimates for an increasing function; trapezoids overestimate for a concave-up function.

Practice questions

76 multiple-choice and 4 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

Water flows into a tank at a rate $r(t)$ liters per hour. The table gives selected values. Using a right Riemann sum with the four subintervals given by the table, what is the approximation of $\displaystyle\int_0^{10}r(t)\,dt$?

1. $79$
2. $70$
3. $\frac{149}{2}$
4. $65$

Question 2 · Multiple choice

What is $\displaystyle\lim_{n\to\infty}\sum_{i=1}^{n}\left(2+\frac{3i}{n}\right)^2\frac{3}{n}$?

1. $9$
2. $117$
3. $39$
4. $13$

Question 3 · Multiple choice

Let $F(x)=\displaystyle\int_1^{x^2}\frac{1}{1+t^3}\,dt$. What is $F'(1)$?

1. $1$
2. $\frac{1}{2}$
3. $2$
4. $0$

Question 4 · Multiple choice

Let $g(x)=\displaystyle\int_0^{x}(t^2-4t+3)\,dt$. Which statement is true?

1. $g$ has a relative minimum at $x=1$ and a relative maximum at $x=3$
2. $g$ has a relative maximum at $x=1$ and a relative minimum at $x=3$
3. $g$ has a relative maximum at $x=2$
4. $g$ is decreasing on $(0,1)$

Question 5 · Multiple choice

If $\displaystyle\int_1^4f(x)\,dx=7$ and $\displaystyle\int_4^6f(x)\,dx=-2$, what is $\displaystyle\int_1^6\big(3f(x)-2\big)\,dx$?

1. $5$
2. $13$
3. $15$
4. $25$

Question 6 · Multiple choice

What is $\displaystyle\int_0^{\pi/3}\left(\sec^2x+\sin x\right)dx$?

1. $- \frac{1}{2} + \sqrt{3}$
2. $\frac{3}{2} + \sqrt{3}$
3. $\frac{1}{2}$
4. $\frac{1}{2} + \sqrt{3}$

Question 7 · Multiple choice

Evaluate $\displaystyle\int_1^{e^2}\frac{(\ln x)^3}{x}\,dx$.

1. $16$
2. $2$
3. $4$
4. $\frac{8}{3}$

Question 8 · Multiple choice

Which of these is $\displaystyle\int\frac{x^2+2x+3}{x+1}\,dx$?

1. $\dfrac{x^2}{2}+x+\ln|x+1|+C$
2. $\dfrac{x^2}{2}+2\ln|x+1|+C$
3. $\dfrac{x^2}{2}+x+2\ln|x+1|+C$
4. $\dfrac{x^3/3+x^2+3x}{x^2/2+x}+C$

Question 9 · Multiple choice

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

1. $\frac{\pi}{4}$
2. $\operatorname{atan}{\left(2 \right)}$
3. $\frac{\log{\left(5 \right)}}{2}$
4. $- \frac{\pi}{4} + \operatorname{atan}{\left(2 \right)}$

Question 10 · Free response (9 points)

Let $f(t)=3-|t-1|$ and let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$ for $-2\le x\le6$.

1. (a) Find $g(4)$ and $g(-2)$. [2 points]
2. (b) Find $g'(3)$ and $g''(3)$. [2 points]
3. (c) Find the absolute maximum value of $g$ on $[-2,6]$. Justify your answer. [3 points]
4. (d) Find the $x$-coordinate of each point of inflection of the graph of $g$. Give a reason. [2 points]

Question 11 · Free response (9 points)

Concert tickets are sold online. The rate of sales is $R(t)$ tickets per hour, where $R$ is differentiable and $t$ is hours after sales open. The table gives selected values of $R(t)$.

1. (a) Use a trapezoidal sum with the five subintervals given by the table to approximate $\displaystyle\int_0^8R(t)\,dt$. Explain the meaning of $\displaystyle\int_0^8R(t)\,dt$ in this context. [3 points]
2. (b) Use your answer to part (a) to approximate the average rate of ticket sales over the first $8$ hours. [1 point]
3. (c) Must there be a time $t$ in $(1,3)$ at which $R'(t)=40$? Justify your answer. [2 points]
4. (d) The rate is modeled by $M(t)=120+70t-9t^2$. Use the model to find the number of tickets sold in the first $8$ hours. [3 points]

Question 12 · Multiple choice

Water flows into a tank at the rate $r(t)=3t+2$ liters per minute for $0\le t\le4$. How many liters of water flow into the tank during these $4$ minutes?

1. $14$
2. $24$
3. $8$
4. $32$

Question 13 · Multiple choice

A cart moves along a straight track. Its velocity $v(t)$, in feet per second, is shown for $0\le t\le6$ seconds. How far does the cart travel during this time?

1. $4$
2. $24$
3. $16$
4. $12$

Question 14 · Multiple choice

The graph shows $T'(t)$, the rate of change of the temperature of a room, in degrees Fahrenheit per hour, for $0\le t\le5$ hours. If $T(0)=50$, what is $T(5)$?

1. $52$
2. $48$
3. $54$
4. $56$

Question 15 · Multiple choice

A colony of bacteria grows at the rate $P'(t)=30e^{0.1t}$ cells per hour. By how many cells does the colony increase from $t=0$ to $t=10$ hours?

1. $81.548$
2. $515.485$
3. $300.000$
4. $815.485$

Question 16 · Multiple choice

Snow falls at a constant rate of $2$ inches per hour for $3$ hours, then at a constant rate of $0.5$ inch per hour for the next $4$ hours. How many inches of snow fall in the $7$ hours?

1. $8$
2. $\frac{5}{2}$
3. $7$
4. $\frac{17}{2}$

Question 17 · Multiple choice

Oil leaks from a drum at the rate $r(t)=6-2t$ gallons per minute for $0\le t\le5$, where a negative rate means oil is being pumped back in. What is the net change in the amount of oil that has left the drum over these $5$ minutes?

1. $9$
2. $-4$
3. $30$
4. $5$

Question 18 · Multiple choice

The function $f$ is continuous; selected values are in the table. What is the left Riemann sum approximation of $\displaystyle\int_0^{10}f(t)\,dt$ using the four subintervals given by the table?

1. $38$
2. $45$
3. $50$
4. $40$

Question 19 · Multiple choice

Selected values of a continuous function $g$ are in the table. What is the right Riemann sum approximation of $\displaystyle\int_0^4g(x)\,dx$ using the three subintervals given by the table?

1. $12$
2. $18$
3. $10$
4. $14$

Question 20 · Multiple choice

A runner's velocity $v(t)$, in meters per second, is recorded at selected times. Using a trapezoidal sum with the three subintervals in the table, what is the approximate distance, in meters, the runner covers for $0\le t\le9$ seconds?

1. $114$
2. $104$
3. $109$
4. $46$

Question 21 · Multiple choice

What is the midpoint Riemann sum approximation of $\displaystyle\int_0^4x^2\,dx$ using two subintervals of equal width?

1. $\frac{64}{3}$
2. $8$
3. $20$
4. $40$

Question 22 · Multiple choice

A left Riemann sum with two equal subintervals is used to approximate $\displaystyle\int_0^1e^x\,dx$. Which gives the approximation and describes it correctly?

1. $\dfrac{1+e^{0.5}}{2}$, an underestimate
2. $\dfrac{1+e^{0.5}}{2}$, an overestimate
3. $\dfrac{e^{0.5}+e}{2}$, an underestimate
4. $\dfrac{e^{0.5}+e}{2}$, an overestimate

Question 23 · Multiple choice

What is the right Riemann sum approximation of $\displaystyle\int_1^3\frac1x\,dx$ using four subintervals of equal width?

1. $\frac{77}{60}$
2. $\ln{\left(3 \right)}$
3. $\frac{19}{20}$
4. $\frac{6}{5}$

Question 24 · Multiple choice

What is $\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\left(1+\frac{2k}{n}\right)^2\frac{2}{n}$?

1. $\frac{26}{3}$
2. $9$
3. $\frac{8}{3}$
4. $\frac{13}{3}$

Question 25 · Multiple choice

Which definite integral is equal to $\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\sqrt{\frac{3k}{n}}\cdot\frac{3}{n}$?

1. $\displaystyle\int_0^3\sqrt{x}\,dx$
2. $\displaystyle\int_0^1\sqrt{3x}\,dx$
3. $\displaystyle\int_0^3\sqrt{3x}\,dx$
4. $\displaystyle\int_1^4\sqrt{x}\,dx$

Question 26 · Multiple choice

Which expression is the right Riemann sum for $\displaystyle\int_0^2x^3\,dx$ with $n$ subintervals of equal width?

1. $\displaystyle\sum_{k=1}^{n}\left(\frac{2k}{n}\right)^3\frac{2}{n}$
2. $\displaystyle\sum_{k=1}^{n}\left(\frac{k}{n}\right)^3\frac{2}{n}$
3. $\displaystyle\sum_{k=1}^{n}\left(\frac{2k}{n}\right)^3\frac{1}{n}$
4. $\displaystyle\sum_{k=0}^{n-1}\left(\frac{2k}{n}\right)^3\frac{2}{n}$

Question 27 · Multiple choice

What is $\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}\frac{k}{n}\cdot\frac{1}{n}$?

1. $1$
2. $\frac{1}{2}$
3. $0$
4. $\infty$

Question 28 · Multiple choice

Interpreting it as an area, find $\displaystyle\int_0^3\sqrt{9-x^2}\,dx$.

1. $\frac{9 \pi}{4}$
2. $\frac{9 \pi}{2}$
3. $3 \pi$
4. $9 \pi$

Question 29 · Multiple choice

What is the value of $\displaystyle\int_0^6|x-2|\,dx$?

1. $6$
2. $8$
3. $10$
4. $12$

Question 30 · Multiple choice

What is the value of $\displaystyle\int_1^5(2x-1)\,dx$, found as the area of a trapezoid?

1. $20$
2. $24$
3. $16$
4. $10$

Question 31 · Multiple choice

Let $g(x)=\displaystyle\int_1^{x}\frac{t^2+1}{t}\,dt$ for $x>0$. What is $g'(2)$?

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

Question 32 · Multiple choice

What is $\dfrac{d}{dx}\displaystyle\int_0^{x^2}\sin t\,dt$ at $x=\sqrt{\pi/2}$?

1. $1$
2. $\sqrt{2} \sqrt{\pi}$
3. $\sqrt{2}$
4. $\pi$

Question 33 · Multiple choice

Let $F(x)=\displaystyle\int_x^{3}e^{t^2}\,dt$. What is $F'(0)$?

1. $1$
2. $0$
3. $-1$
4. $e^{9}$

Question 34 · Multiple choice

Let $h(x)=\displaystyle\int_2^{3x}\sqrt{1+t^3}\,dt$. What is $h'(1)$?

1. $6 \sqrt{7}$
2. $2 \sqrt{7}$
3. $3$
4. $\frac{2 \sqrt{7}}{3}$

Question 35 · Multiple choice

The graph of $f$, made of two line segments and a semicircle, is shown. Let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$ for $0\le x\le8$. On which set is $g$ increasing?

1. $(4,8)$
2. $(0,2)\cup(6,8)$
3. $(2,6)$
4. $(0,4)$

Question 36 · Multiple choice

The graph of $f$, made of two line segments and a semicircle, is shown. Let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$ for $0\le x\le8$. At which value of $x$ does $g$ have a relative minimum?

1. $6$
2. $2$
3. $4$
4. $8$

Question 37 · Multiple choice

The graph of $f$, made of two line segments and a semicircle, is shown. Let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$ for $0\le x\le8$. What is $g(4)$?

1. $2 + \pi$
2. $2 - 2 \pi$
3. $2 - \pi$
4. $- \pi$

Question 38 · Multiple choice

The graph of $f$, made of two line segments and a semicircle, is shown. Let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$ for $0\le x\le8$. What is the absolute minimum value of $g$ on $[0,8]$?

1. $0$
2. $4 - 2 \pi$
3. $2 - 2 \pi$
4. $- 2 \pi$

Question 39 · Multiple choice

The graph of $f$, made of two line segments and a semicircle, is shown. Let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$ for $0\le x\le8$. On which interval is the graph of $g$ concave up?

1. $(6,8)$
2. $(0,4)$
3. $(4,8)$
4. $(2,6)$

Question 40 · Multiple choice

If $\displaystyle\int_1^5f(x)\,dx=8$ and $\displaystyle\int_1^3f(x)\,dx=3$, what is $\displaystyle\int_5^3 2f(x)\,dx$?

1. $-10$
2. $10$
3. $-5$
4. $22$

Question 41 · Multiple choice

If $\displaystyle\int_0^4f(x)\,dx=6$, what is $\displaystyle\int_0^4\big(3f(x)-2\big)\,dx$?

1. $10$
2. $16$
3. $-2$
4. $12$

Question 42 · Multiple choice

What is $\displaystyle\int_{-\pi}^{\pi}x^3\cos x\,dx$?

1. $0$
2. $2 \pi$
3. $- 12 \pi$
4. $\frac{\pi^{4}}{2}$

Question 43 · Multiple choice

Let $f(x)=\begin{cases}2, & x<1\\ x+1, & x\ge1.\end{cases}$ What is $\displaystyle\int_{-1}^{3}f(x)\,dx$?

1. $8$
2. $12$
3. $10$
4. $6$

Question 44 · Multiple choice

What is $\displaystyle\int_1^4\left(3\sqrt{x}-\frac{1}{x^2}\right)dx$?

1. $14$
2. $\frac{55}{4}$
3. $\frac{53}{4}$
4. $\frac{51}{4}$

Question 45 · Multiple choice

What is $\displaystyle\int_0^{\pi/3}\sec^2x\,dx$?

1. $3$
2. $\sqrt{3}$
3. $\frac{\sqrt{3}}{3}$
4. $2$

Question 46 · Multiple choice

What is $\displaystyle\int_1^{e}\frac{2}{x}\,dx$?

1. $1$
2. $-2 + 2 e$
3. $2$
4. $e$

Question 47 · Multiple choice

What is $\displaystyle\int_0^{\ln 3}e^x\,dx$?

1. $2$
2. $3$
3. $\ln{\left(3 \right)}$
4. $1$

Question 48 · Multiple choice

What is $\displaystyle\int_0^1(x+1)(x-2)\,dx$?

1. $- \frac{13}{6}$
2. $-2$
3. $\frac{13}{6}$
4. $- \frac{7}{6}$

Question 49 · Multiple choice

What is $\displaystyle\int_0^{1/2}\frac{1}{\sqrt{1-x^2}}\,dx$?

1. $\frac{\pi}{3}$
2. $\frac{1}{2}$
3. $\frac{\pi}{6}$
4. $\frac{\pi}{4}$

Question 50 · Multiple choice

Which of the following is $\displaystyle\int(8x^3-9x^2+4)\,dx$? ($C$ is a constant.)

1. $24 x^{2} - 18 x+C$
2. $2 x^{4} - 3 x^{3} + 4 x+C$
3. $2 x^{4} - 9 x^{3} + 4 x+C$
4. $8 x^{4} - 3 x^{3} + 4 x+C$

Question 51 · Multiple choice

Which of the following is $\displaystyle\int\left(\frac1x+e^x\right)dx$? ($C$ is a constant.)

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

Question 52 · Multiple choice

Which of the following is $\displaystyle\int(\cos x-3\sin x)\,dx$? ($C$ is a constant.)

1. $\sin{\left(x \right)} + 3 \cos{\left(x \right)}+C$
2. $\sin{\left(x \right)} - 3 \cos{\left(x \right)}+C$
3. $- \sin{\left(x \right)} + 3 \cos{\left(x \right)}+C$
4. $- \sin{\left(x \right)} - 3 \cos{\left(x \right)}+C$

Question 53 · Multiple choice

Which of the following is $\displaystyle\int\sec x\tan x\,dx$? ($C$ is a constant.)

1. $\tan{\left(x \right)}+C$
2. $\sec{\left(x \right)}+C$
3. $\frac{\sec^{2}{\left(x \right)}}{2}+C$
4. $- \sec{\left(x \right)}+C$

Question 54 · Multiple choice

Which of the following is $\displaystyle\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)dx$? ($C$ is a constant.)

1. $\frac{3 x^{\frac{3}{2}}}{2} + 2 \sqrt{x}+C$
2. $\frac{2 x^{\frac{3}{2}}}{3} + \frac{\sqrt{x}}{2}+C$
3. $x^{\frac{3}{2}} + \sqrt{x}+C$
4. $\frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x}+C$

Question 55 · Multiple choice

A function $f$ has derivative $f'(x)=9x^2-2x$, and its graph passes through $(1,4)$. What is $f(2)$?

1. $20$
2. $34$
3. $18$
4. $22$

Question 56 · Multiple choice

Which of the following is $\displaystyle\int2x(x^2+1)^4\,dx$? ($C$ is a constant.)

1. $\left(x^{2} + 1\right)^{5}+C$
2. $\frac{2 x \left(x^{2} + 1\right)^{5}}{5}+C$
3. $\frac{\left(x^{2} + 1\right)^{5}}{5}+C$
4. $\frac{x^{2} \left(x^{2} + 1\right)^{5}}{5}+C$

Question 57 · Multiple choice

What is $\displaystyle\int_0^1xe^{x^2}\,dx$?

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

Question 58 · Multiple choice

What is $\displaystyle\int_0^{\pi/2}\sin^2x\cos x\,dx$?

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{2}{3}$
4. $1$

Question 59 · Multiple choice

For $x>0$, which of the following is $\displaystyle\int\frac{(\ln x)^2}{x}\,dx$? ($C$ is a constant.)

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

Question 60 · Multiple choice

What is $\displaystyle\int_0^2\frac{x}{\sqrt{x^2+5}}\,dx$?

1. $3$
2. $\frac{3}{2} - \frac{\sqrt{5}}{2}$
3. $6 - 2 \sqrt{5}$
4. $3 - \sqrt{5}$

Question 61 · Multiple choice

What is $\displaystyle\int_1^2\frac{2x}{x^2+1}\,dx$?

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

Question 62 · Multiple choice

Which of the following is $\displaystyle\int\frac{x^2+1}{x+1}\,dx$? ($C$ is a constant.)

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

Question 63 · Multiple choice

What is $\displaystyle\int_0^1\frac{x^2}{x+1}\,dx$?

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

Question 64 · Multiple choice

Which of the following is $\displaystyle\int\frac{1}{x^2+2x+5}\,dx$? ($C$ is a constant.)

1. $\operatorname{atan}{\left(x + 1 \right)}+C$
2. $\operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)}+C$
3. $\frac{\operatorname{atan}{\left(\frac{x}{2} + \frac{1}{2} \right)}}{2}+C$
4. $\ln{\left(x^{2} + 2 x + 5 \right)}+C$

Question 65 · Multiple choice

What is $\displaystyle\int_1^3\frac{1}{x^2-2x+5}\,dx$?

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

Question 66 · Multiple choice

Which of the following is $\displaystyle\int\frac{x}{x^2+4}\,dx$? ($C$ is a constant.)

1. $\frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}+C$
2. $\frac{\ln{\left(x^{2} + 4 \right)}}{2}+C$
3. $\ln{\left(x^{2} + 4 \right)}+C$
4. $\frac{x^{2}}{2 x^{2} + 8}+C$

Question 67 · Multiple choice

Which of the following is $\displaystyle\int\frac{1}{x^2+4}\,dx$? ($C$ is a constant.)

1. $\frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}+C$
2. $\frac{\ln{\left(x^{2} + 4 \right)}}{2}+C$
3. $\operatorname{atan}{\left(\frac{x}{2} \right)}+C$
4. $\frac{\operatorname{atan}{\left(x \right)}}{4}+C$

Question 68 · Multiple choice

Which of the following is $\displaystyle\int\frac{x+1}{x}\,dx$? ($C$ is a constant.)

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

Question 69 · Multiple choice

Which of the following is $\displaystyle\int\tan x\,dx$? ($C$ is a constant.)

1. $\ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}+C$
2. $\sec^{2}{\left(x \right)}+C$
3. $- \ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}+C$
4. $\frac{\tan^{2}{\left(x \right)}}{2}+C$

Question 70 · Multiple choice

What is $\displaystyle\int_0^{\pi/4}\tan^2x\,dx$?

1. $1$
2. $1 - \frac{\pi}{4}$
3. $\frac{\pi}{4}$
4. $\frac{1}{3}$

Question 71 · Multiple choice

Which of the following is $\displaystyle\int\frac{e^{2x}}{1+e^{2x}}\,dx$? ($C$ is a constant.)

1. $\ln{\left(e^{2 x} + 1 \right)}+C$
2. $\frac{\ln{\left(e^{2 x} + 1 \right)}}{2}+C$
3. $\operatorname{atan}{\left(e^{x} \right)}+C$
4. $\frac{e^{2 x}}{2}+C$

Question 72 · Multiple choice

Which of the following is $\displaystyle\int\frac{1}{\sqrt{9-x^2}}\,dx$? ($C$ is a constant.)

1. $\operatorname{asin}{\left(\frac{x}{3} \right)}+C$
2. $\frac{\operatorname{asin}{\left(\frac{x}{3} \right)}}{3}+C$
3. $3 \operatorname{asin}{\left(x \right)}+C$
4. $\frac{\operatorname{atan}{\left(\frac{x}{3} \right)}}{3}+C$

Question 73 · Multiple choice

Which of the following is $\displaystyle\int(3x-2)^5\,dx$? ($C$ is a constant.)

1. $\frac{\left(3 x - 2\right)^{6}}{18}+C$
2. $\frac{\left(3 x - 2\right)^{6}}{6}+C$
3. $\frac{\left(3 x - 2\right)^{6}}{2}+C$
4. $5 \left(3 x - 2\right)^{4}+C$

Question 74 · Multiple choice

The graph of $f$, made of two line segments and a semicircle, is shown. Let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$ for $0\le x\le8$. What is $\displaystyle\int_8^0f(t)\,dt$?

1. $4 - 2 \pi$
2. $-4 + 2 \pi$
3. $4 + 2 \pi$
4. $2 \pi$

Question 75 · Multiple choice

Which of the following is $\displaystyle\int\frac{x^2-1}{x^2}\,dx$? ($C$ is a constant.)

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

Question 76 · Multiple choice

What is $\displaystyle\int_1^4\frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx$?

1. $- e + e^{2}$
2. $2 e^{2}$
3. $- e + e^{4}$
4. $- 2 e + 2 e^{2}$

Question 77 · Multiple choice

Which of the following is $\displaystyle\int\frac{x^3}{x^2+1}\,dx$? ($C$ is a constant.)

1. $\frac{x^{2}}{2} + \frac{\ln{\left(x^{2} + 1 \right)}}{2}+C$
2. $\frac{x^{2}}{2} - \frac{\ln{\left(x^{2} + 1 \right)}}{2}+C$
3. $\frac{x^{2}}{2} - \operatorname{atan}{\left(x \right)}+C$
4. $\frac{x^{4}}{\frac{4 x^{3}}{3} + 4 x}+C$

Question 78 · Multiple choice

What is $\displaystyle\int_1^2\frac{1}{x^2-2x+2}\,dx$?

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

Question 79 · Free response (9 points)

The graph of the continuous function $f$, made of three line segments, is shown for $-2\le x\le6$. Let $g(x)=\displaystyle\int_0^{x}f(t)\,dt$.

1. (a) Find $g(2)$ and $g(-2)$. [2 points]
2. (b) Find $g'(1)$ and $g''(-1)$. [2 points]
3. (c) At what value of $x$ in $(-2,6)$ does $g$ have a relative maximum? Justify your answer. [2 points]
4. (d) Find the absolute minimum value of $g$ on $[-2,6]$. Justify your answer. [3 points]

Question 80 · Free response (9 points)

Water flows into a pond at a rate of $R(t)$ gallons per hour, where $R$ is continuous and $t$ is measured in hours. Selected values of $R(t)$ are given in the table.

1. (a) Use a trapezoidal sum with the four subintervals in the table to approximate $\displaystyle\int_0^{10}R(t)\,dt$. Indicate units of measure, and explain the meaning of the integral in context. [3 points]
2. (b) Use a left Riemann sum with the four subintervals in the table to approximate $\displaystyle\int_0^{10}R(t)\,dt$. [2 points]
3. (c) A model for the rate is $W(t)=12+2.4t-0.3t^2$. Use the Fundamental Theorem of Calculus to find $\displaystyle\int_0^{10}W(t)\,dt$. [2 points]
4. (d) Find the value of $\displaystyle\lim_{n\to\infty}\sum_{k=1}^{n}W\!\left(\frac{5k}{n}\right)\frac{5}{n}$. [2 points]