This practice page is modeled after the exam topics from the uploaded test. It includes the original-style problem set plus 40 additional practice problems that are similar in style but not identical.
Problems 1-5: Work each problem.
Problems 6-10: These are integration and improper integral style problems modeled after the original exam.
Extra Practice 11-50: Similar topics, but not exact repeats. Difficulty varies.
Formula Sheet: Included at the bottom of this page for quick review.
Compute \(\dfrac{dy}{dx}\) for each:
(a) \(y = 6x + \ln(2x+5) + \tan^{-1}(\cosh x)\)
(b) \(y = \sin(x^x)\)
Difficulty: varies, about 3/5 to 4/5Compute the following limits if they exist:
(a) \(\displaystyle \lim_{x\to 0} \frac{1-\cos(3x)}{2x^2+x-\sin x}\)
(b) Find the value of \(A\) so that L'Hospital's Rule applies, then compute
\(\displaystyle \lim_{x\to 3} \frac{x^2-2x+A}{2x-6}\)
Difficulty: about 3/5Compute each:
(a) \(\sin(\arcsin(-1))\)
(b) \(\arcsin(\sin(2\pi))\)
(c) \(\arctan(\cos \pi)\)
(d) \(\arcsin\!\left(\cos\!\left(\frac{\pi}{3}\right)\right)\)
(e) \(\arcsin(\cos \pi)\)
Difficulty: about 2/5 to 3/5A bacteria population grows according to \(P(t)=200e^{2t}\), where \(t\) is in days.
Find the initial population, when the population reaches 1000, and the rate of change after 3 days.
Difficulty: about 2/5 to 3/5Complete the identity using the triangle method:
\(\cos(\sin^{-1}(3x))\)
Difficulty: about 3/5Evaluate:
\(\displaystyle \int x^2\cos(3x)\,dx\)
Difficulty: about 4/5Evaluate:
\(\displaystyle \int \cos^6(x)\sin^3(x)\,dx\)
Difficulty: about 4/5Evaluate:
\(\displaystyle \int \frac{x^2}{\sqrt{9-x^2}}\,dx\)
Difficulty: about 3/5Evaluate:
\(\displaystyle \int \frac{5x-2}{x(x^2+1)}\,dx\)
Difficulty: about 3/5Determine whether the integral converges or diverges. If it converges, find its value:
\(\displaystyle \int_0^{\infty} \frac{x^4}{(1+x^5)^2}\,dx\)
Difficulty: about 4/5 to 5/5Compute \(\dfrac{dy}{dx}\): \(y = 4x + \ln(5x-1) + \tan^{-1}(\sinh x)\)
Difficulty: 3/5Compute \(\dfrac{dy}{dx}\): \(y = \cos(x^x)\)
Difficulty: 4/5Compute \(\displaystyle \lim_{x\to 0} \frac{1-\cos(4x)}{x^2}\)
Difficulty: 2/5Find \(A\) so that L'Hospital's Rule applies, then compute \(\displaystyle \lim_{x\to 2} \frac{x^2-5x+A}{x-2}\).
Difficulty: 3/5Compute: \(\sin(\arcsin(1/2))\)
Difficulty: 2/5Compute: \(\arcsin(\sin(5\pi/6))\)
Difficulty: 3/5Compute: \(\arctan(\cos 0)\)
Difficulty: 2/5Use the triangle method to simplify \(\sin(\tan^{-1}(2x))\).
Difficulty: 3/5A population grows according to \(P(t)=350e^{1.5t}\). Find the initial population and when it reaches 2000.
Difficulty: 2/5 to 3/5For \(P(t)=120e^{0.8t}\), find \(P'(t)\) and then compute \(P'(4)\).
Difficulty: 2/5Evaluate: \(\displaystyle \int x e^{2x}\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int x^2 \sin(2x)\,dx\)
Difficulty: 4/5Evaluate: \(\displaystyle \int x\ln x\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int e^x\cos x\,dx\)
Difficulty: 4/5Evaluate: \(\displaystyle \int \sin^3(x)\cos^4(x)\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int \cos^5(x)\sin^2(x)\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int \sin^2(x)\cos^2(x)\,dx\)
Difficulty: 4/5Evaluate: \(\displaystyle \int \sec^3(x)\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int \sec(x)\tan(x)\,dx\)
Difficulty: 2/5Evaluate: \(\displaystyle \int \csc(x)\cot(x)\,dx\)
Difficulty: 2/5Evaluate: \(\displaystyle \int \frac{dx}{\sqrt{16-x^2}}\)
Difficulty: 3/5Evaluate: \(\displaystyle \int \frac{x^2}{\sqrt{25-x^2}}\,dx\)
Difficulty: 3/5 to 4/5Evaluate: \(\displaystyle \int \frac{dx}{x\sqrt{x^2-9}}\)
Difficulty: 4/5Evaluate: \(\displaystyle \int \sqrt{9-x^2}\,dx\)
Difficulty: 4/5Evaluate: \(\displaystyle \int \frac{3x+5}{x(x+2)}\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int \frac{2x+7}{x^2+x-6}\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int \frac{x^2+1}{x(x-1)(x+1)}\,dx\)
Difficulty: 4/5Evaluate: \(\displaystyle \int \frac{5}{x^2-4}\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int \frac{4x-1}{x(x^2+4)}\,dx\)
Difficulty: 4/5Evaluate: \(\displaystyle \int \frac{dx}{x^2-9}\)
Difficulty: 3/5Determine whether \(\displaystyle \int_1^{\infty} \frac{1}{x^2}\,dx\) converges or diverges. If it converges, find its value.
Difficulty: 2/5Determine whether \(\displaystyle \int_1^{\infty} \frac{1}{x}\,dx\) converges or diverges.
Difficulty: 2/5Determine whether \(\displaystyle \int_0^{1} \frac{1}{\sqrt{x}}\,dx\) converges or diverges. If it converges, find its value.
Difficulty: 2/5Determine whether \(\displaystyle \int_0^{\infty} e^{-x}\,dx\) converges or diverges. If it converges, find its value.
Difficulty: 2/5Determine whether \(\displaystyle \int_1^{\infty} \frac{1}{x^{3/2}}\,dx\) converges or diverges. If it converges, find its value.
Difficulty: 3/5Determine whether \(\displaystyle \int_1^{\infty} \frac{dx}{x^{4/5}}\) converges or diverges.
Difficulty: 3/5Evaluate: \(\displaystyle \int_0^{\infty} \frac{3}{(1+3x)^2}\,dx\)
Difficulty: 3/5Evaluate: \(\displaystyle \int_2^{\infty} \frac{1}{x\ln x}\,dx\) or determine divergence.
Difficulty: 4/5Evaluate: \(\displaystyle \int \cos(\ln x)\,dx\)
Difficulty: 4/5Evaluate: \(\displaystyle \int \sin(\ln(2x))\,dx\)
Difficulty: 4/5Use the dropdowns below only after attempting the problems yourself.
1(a) \(y' = 6 + \dfrac{2}{2x+5} + \dfrac{\sinh x}{1+\cosh^2 x}\)
1(b) \(y' = \cos(x^x)\,x^x(\ln x + 1)\)
2(a) \(\dfrac{9}{2}\)
2(b) \(A=-3\), and the limit is \(2\)
3(a) \(-1\)
3(b) \(0\)
3(c) \(-\pi/4\)
3(d) \(\pi/6\)
3(e) does not exist in the real numbers since \(\arcsin(-1)= -\pi/2\) but here the input is \(\cos\pi=-1\), so the value is actually \(-\pi/2\)
4 Initial population \(=200\). Time to reach 1000: \(t=\frac{\ln 5}{2}\). Rate after 3 days: \(P'(3)=400e^6\)
5 \(\sqrt{1-9x^2}\)
6 \(\displaystyle \frac{x^2\sin(3x)}{3}+\frac{2x\cos(3x)}{9}-\frac{2\sin(3x)}{27}+C\)
7 \(\displaystyle \frac{\cos^9 x}{9}-\frac{2\cos^{11}x}{11}+\frac{\cos^{13}x}{13}+C\)
8 \(\displaystyle \frac{9}{2}\arcsin\!\left(\frac{x}{3}\right)-\frac{x}{2}\sqrt{9-x^2}+C\)
9 \(\displaystyle -2\ln|x|+\frac{9}{2}\ln(x^2+1)-2\arctan(x)+C\)
10 Converges to \(\dfrac{1}{5}\)
11 \(y' = 4 + \dfrac{5}{5x-1} + \dfrac{\cosh x}{1+\sinh^2 x}\)
12 \(y' = -\sin(x^x)\,x^x(\ln x+1)\)
13 \(8\)
14 \(A=6\), limit \(=-1\)
15 \(1/2\)
16 \(\pi/6\)
17 \(\pi/4\)
18 \(\dfrac{2x}{\sqrt{1+4x^2}}\)
19 Initial population \(=350\), reaching 2000 at \(t=\dfrac{1}{1.5}\ln\!\left(\dfrac{40}{7}\right)\)
20 \(P'(t)=96e^{0.8t}\), so \(P'(4)=96e^{3.2}\)
21 \(\displaystyle \frac{1}{2}xe^{2x}-\frac{1}{4}e^{2x}+C\)
22 \(\displaystyle -\frac{x^2\cos(2x)}{2}+\frac{x\sin(2x)}{2}+\frac{\cos(2x)}{4}+C\)
23 \(\displaystyle \frac{x^2}{2}\ln x-\frac{x^2}{4}+C\)
24 \(\displaystyle \frac{e^x(\sin x+\cos x)}{2}+C\)
25 \(\displaystyle \frac{\cos^7 x}{7}-\frac{\cos^5 x}{5}+C\)
26 \(\displaystyle \frac{\sin^3 x}{3}-\frac{2\sin^5 x}{5}+\frac{\sin^7 x}{7}+C\)
27 \(\displaystyle \frac{x}{8}-\frac{\sin(4x)}{32}+C\)
28 \(\displaystyle \frac12\sec x\tan x+\frac12\ln|\sec x+\tan x|+C\)
29 \(\sec x + C\)
30 \(-\csc x + C\)
31 \(\arcsin(x/4)+C\)
32 \(\displaystyle \frac{25}{2}\arcsin\!\left(\frac{x}{5}\right)-\frac{x}{2}\sqrt{25-x^2}+C\)
33 \(\displaystyle \frac{1}{3}\sec^{-1}\!\left(\frac{|x|}{3}\right)+C\)
34 \(\displaystyle \frac{x}{2}\sqrt{9-x^2}+\frac{9}{2}\arcsin\!\left(\frac{x}{3}\right)+C\)
35 \(\displaystyle \frac{5}{2}\ln|x|+\frac{1}{2}\ln|x+2|+C\)
36 \(\displaystyle \frac{11}{5}\ln|x-2|-\frac{1}{5}\ln|x+3|+C\)
37 \(\ln|x|-\frac{1}{x-1}+\frac{1}{x+1}+C\)
38 \(\displaystyle \frac{5}{4}\ln\left|\frac{x-2}{x+2}\right|+C\)
39 \(\displaystyle -\frac14\ln|x|+\frac18\ln(x^2+4)+\frac72\arctan(x/2)+C\)
40 \(\displaystyle \frac{1}{6}\ln\left|\frac{x-3}{x+3}\right|+C\)
41 Converges to \(1\)
42 Diverges
43 Converges to \(2\)
44 Converges to \(1\)
45 Converges to \(2\)
46 Diverges
47 Converges to \(1\)
48 Diverges
49 \(\displaystyle \frac{x}{2}(\sin(\ln x)+\cos(\ln x))+C\)
50 \(\displaystyle \frac{x}{2}(\sin(\ln(2x))-\cos(\ln(2x)))+C\)
Use the dropdowns below only after attempting the problems yourself.
1(a) \\(y' = 6 + \\dfrac{2}{2x+5} + \\dfrac{\\sinh x}{1+\\cosh^2 x}\\)
1(b) \\(y' = \\cos(x^x)\\,x^x(\\ln x + 1)\\)
2(a) \\(\\dfrac{9}{2}\\)
2(b) \\(A=-3\\), and the limit is \\(2\\)
3(a) \\(-1\\)
3(b) \\(0\\)
3(c) \\(-\\pi/4\\)
3(d) \\(\\pi/6\\)
3(e) \\(-\\pi/2\\)
4 Initial population \\(=200\\). Time to reach 1000: \\(t=\\frac{\\ln 5}{2}\\). Rate after 3 days: \\(P'(3)=400e^6\\)
5 \\(\\sqrt{1-9x^2}\\)
6 \\(\\displaystyle \\frac{x^2\\sin(3x)}{3}+\\frac{2x\\cos(3x)}{9}-\\frac{2\\sin(3x)}{27}+C\\)
7 \\(\\displaystyle \\frac{\\cos^7 x}{7}-\\frac{2\\cos^9 x}{9}+\\frac{\\cos^{11}x}{11}+C\\)
8 \\(\\displaystyle \\frac{9}{2}\\arcsin\\!\\left(\\frac{x}{3}\\right)-\\frac{x}{2}\\sqrt{9-x^2}+C\\)
9 \\(\\displaystyle -2\\ln|x|+\\frac{9}{2}\\ln(x^2+1)-2\\arctan(x)+C\\)
10 Converges to \\(\\dfrac{1}{5}\\)
11 \\(y' = 4 + \\dfrac{5}{5x-1} + \\dfrac{\\cosh x}{1+\\sinh^2 x}\\)
12 \\(y' = -\\sin(x^x)\\,x^x(\\ln x+1)\\)
13 \\(8\\)
14 \\(A=6\\), limit \\(=-1\\)
15 \\(1/2\\)
16 \\(\\pi/6\\)
17 \\(\\pi/4\\)
18 \\(\\dfrac{2x}{\\sqrt{1+4x^2}}\\)
19 Initial population \\(=350\\), reaching 2000 at \\(t=\\dfrac{1}{1.5}\\ln\\!\\left(\\dfrac{40}{7}\\right)\\)
20 \\(P'(t)=96e^{0.8t}\\), so \\(P'(4)=96e^{3.2}\\)
21 \\(\\displaystyle \\frac{1}{2}xe^{2x}-\\frac{1}{4}e^{2x}+C\\)
22 \\(\\displaystyle -\\frac{x^2\\cos(2x)}{2}+\\frac{x\\sin(2x)}{2}+\\frac{\\cos(2x)}{4}+C\\)
23 \\(\\displaystyle \\frac{x^2}{2}\\ln x-\\frac{x^2}{4}+C\\)
24 \\(\\displaystyle \\frac{e^x(\\sin x+\\cos x)}{2}+C\\)
25 \\(\\displaystyle \\frac{\\cos^5 x}{5}-\\frac{\\cos^7 x}{7}+C\\)
26 \\(\\displaystyle \\frac{\\sin^3 x}{3}-\\frac{2\\sin^5 x}{5}+\\frac{\\sin^7 x}{7}+C\\)
27 \\(\\displaystyle \\frac{x}{8}-\\frac{\\sin(4x)}{32}+C\\)
28 \\(\\displaystyle \\frac12\\sec x\\tan x+\\frac12\\ln|\\sec x+\\tan x|+C\\)
29 \\(\\sec x + C\\)
30 \\(-\\csc x + C\\)
31 \\(\\arcsin(x/4)+C\\)
32 \\(\\displaystyle \\frac{25}{2}\\arcsin\\!\\left(\\frac{x}{5}\\right)-\\frac{x}{2}\\sqrt{25-x^2}+C\\)
33 \\(\\displaystyle \\frac{1}{3}\\sec^{-1}\\!\\left(\\frac{|x|}{3}\\right)+C\\)
34 \\(\\displaystyle \\frac{x}{2}\\sqrt{9-x^2}+\\frac{9}{2}\\arcsin\\!\\left(\\frac{x}{3}\\right)+C\\)
35 \\(\\displaystyle \\frac{5}{2}\\ln|x|+\\frac{1}{2}\\ln|x+2|+C\\)
36 \\(\\displaystyle \\frac{11}{5}\\ln|x-2|-\\frac{1}{5}\\ln|x+3|+C\\)
37 \\(\\ln|x|-\\frac{1}{x-1}+\\frac{1}{x+1}+C\\)
38 \\(\\displaystyle \\frac{5}{4}\\ln\\left|\\frac{x-2}{x+2}\\right|+C\\)
39 \\(\\displaystyle -\\frac14\\ln|x|+\\frac18\\ln(x^2+4)+\\frac72\\arctan(x/2)+C\\)
40 \\(\\displaystyle \\frac{1}{6}\\ln\\left|\\frac{x-3}{x+3}\\right|+C\\)
41 Converges to \\(1\\)
42 Diverges
43 Converges to \\(2\\)
44 Converges to \\(1\\)
45 Converges to \\(2\\)
46 Diverges
47 Converges to \\(1\\)
48 Diverges
49 \\(\\displaystyle \\frac{x}{2}(\\sin(\\ln x)+\\cos(\\ln x))+C\\)
50 \\(\\displaystyle \\frac{x}{2}(\\sin(\\ln(2x))-\\cos(\\ln(2x)))+C\\)