Topic 1: Introduction to Integration and Antiderivatives
1\( x^2 + C \)
2\( 5x + C \)
3\( \frac{x^3}{3} + C \)
4\( x^3 + C \)
5\( x + C \)
6\( \frac{x^2}{2} + C \)
7\( x^4 + C \)
8\( \frac{x^4}{2} + C \)
9\( \frac{x^2}{2} + x + C \)
10\( \frac{x^3}{3} + 2x + C \)
11\( -\frac{1}{x} + C \)
12\( x^3 - x^2 + x + C \)
13\( \frac{2}{3} x^{3/2} + C \)
14\( \frac{x^4}{4} - \frac{x^2}{2} + C \)
15\( 2 \ln |x| + C \)
16\( \frac{4x^3}{3} + \frac{3x^2}{2} - 2x + C \)
17\( 2x^{1/2} + C \)
18\( \frac{x^5}{5} - \frac{2x^3}{3} + \frac{x^2}{2} + C \)
19\( x^5 - x^3 + C \)
20\( -\frac{1}{x} + \frac{x^2}{2} + C \)
21\( -\frac{1}{2x^2} + \frac{1}{x} + C \)
22\( e^x - \cos x + C \)
23\( x^2 e^x - 2x e^x + 2 e^x + C \)
24\( \sin x + \cos x + C \)
25\( \arctan x + C \)
Topic 2: Indefinite Integrals and Basic Integration Rules
1\( 3x^2 + C \)
2\( \frac{x^4}{4} + C \)
3\( 4x + C \)
4\( \frac{2x^3}{3} + C \)
5\( \frac{x^2}{2} + 3x + C \)
6\( x^5 + C \)
7\( x^2 - x + C \)
8\( \frac{x^6}{6} + C \)
9\( x^3 + x + C \)
10\( \frac{7x^2}{2} + C \)
11\( \frac{x^3}{3} - 2x^2 + 2x + C \)
12\( -\frac{1}{2x^2} + C \)
13\( \frac{4}{3} x^{3/2} - \frac{x^2}{2} + C \)
14\( \frac{3}{2} x^{2/3} + x^2 + C \)
15\( x^3 - \frac{5x^2}{2} + 4x + C \)
16\( -\frac{2}{x} + C \)
17\( \frac{2}{3} x^{3/2} + 2x^{1/2} + C \)
18\( \frac{x^4}{4} - x^2 + 3x + C \)
19\( -2x^{-2} + \frac{x^3}{3} + C \)
20\( x^5 - \frac{x^4}{2} + \frac{x^2}{2} + C \)
21\( \frac{e^{2x}}{2} - \frac{\sin 2x}{2} + C \)
22\( \frac{x^3 \ln x}{3} - \frac{x^3}{9} + C \)
23\( -\frac{\cos 3x}{3} - \frac{\sin 2x}{2} + C \)
24\( -x e^{-x} - e^{-x} + C \)
25\( \frac{(\ln x)^2}{2} + C \)
Topic 3: Integration by Substitution (u-Substitution)
1\( e^{x^2} + C \)
2\( \frac{e^{x^3}}{3} + C \)
3\( -\frac{\cos^3 x}{3} + C \)
4\( \frac{1}{2} \ln |x^2 + 1| + C \)
5\( \frac{1}{2} e^{2x} + C \)
6\( \frac{1}{2} e^{x^2 + 1} + C \)
7\( -\frac{\cos^4 x}{4} + C \)
8\( \frac{1}{2} \ln |2x + 1| + C \)
9\( \frac{2}{3} (x^4 + 1)^{3/2} + C \)
10\( -\cos(e^{-x}) + C \)
11\( \frac{2}{3} (x^3 + 1)^{3/2} + C \)
12\( -2 \cos \sqrt{x} + C \)
13\( \frac{1}{3} e^{x^3 - 2} + C \)
14\( -\frac{\cos^3 x}{3} + C \)
15\( -\frac{1}{2(x^2 + 3)} + C \)
16\( \frac{1}{4} \sin(2x^2) + C \)
17\( 2 e^{\sqrt{x}} + C \)
18\( \sin(\ln x) + C \)
19\( -\frac{1}{3} e^{-x^3} + C \)
20\( -\frac{1}{\cos x} + C \)
21\( -\frac{2}{3} (1 - x^4)^{3/2} + C \)
22\( \frac{2}{3} (1 + \sin^4 x)^{3/2} + C \)
23\( -\cos(e^{x^3}) + C \)
24\( \ln |1 + e^{\tan x}| + C \)
25\( \frac{3}{4} (x^3 + 3x)^{4/3} + C \)
Topic 4: Definite Integrals and the Fundamental Theorem of Calculus
1\( 4 \)
2\( \frac{3}{2} \)
3\( 4 \)
4\( 9 \)
5\( \frac{26}{3} \)
6\( 5 \)
7\( \frac{5}{4} \)
8\( 60 \)
9\( \frac{14}{3} \)
10\( \frac{93}{5} \)
11\( -\frac{1}{3} \)
12\( \frac{1}{2} \)
13\( \frac{2}{3} \)
14\( \frac{2}{3} \)
15\( 4 \)
16\( 2 \)
17\( \frac{14}{3} \)
18\( 2 \ln 4 \)
19\( -\frac{1}{12} \)
20\( -\frac{1}{2} + \frac{13}{6} \)
21\( e - 1 \)
22\( 2 \)
23\( \frac{3}{4} \ln 2 - \frac{1}{4} \)
24\( \frac{\pi}{4} \)
25\( \frac{\pi}{4} \)
Topic 5: Area Under a Curve and Between Two Curves
1\( 2 \)
2\( \frac{1}{3} \)
3\( 9 \)
4\( 9 \)
5\( \frac{4}{3} \)
6\( 6 \)
7\( \frac{4}{3} \)
8\( \frac{16}{3} \)
9\( \frac{16}{3} \)
10\( \ln 2 \)
11\( \frac{1}{6} \)
12\( \frac{8}{3} \)
13\( \frac{2}{3} \)
14\( 2 \)
15\( \frac{9}{2} \)
16\( \frac{1}{6} \)
17\( \frac{2}{3} - \frac{1}{4} \)
18\( 1 \)
19\( \frac{2}{3} \)
20\( \ln 2 - \frac{3}{4} \)
21\( e - \frac{1}{e} \)
22\( \frac{\pi}{2} \)
23\( \frac{5}{4} \)
24\( \frac{\pi}{4} - \frac{1}{2} \)
25\( \frac{\pi}{2} + \frac{2}{3} \)
Topic 6: Applications of Integration (Accumulated Change, Average Value)
1\( 9 \)
2\( 6 \)
3\( 4 \)
4\( \frac{1}{3} \)
5\( 1 \)
6\( 2 \)
7\( 9 \)
8\( \frac{1}{3} \)
9\( \frac{16}{3} \)
10\( 3 \)
11\( 4 \)
12\( \frac{4}{3} \)
13\( 2 \)
14\( e - 1 \)
15\( 2 \)
16\( 0 \)
17\( \ln 2 \)
18\( \frac{2}{\pi} \)
19\( \frac{10}{3} \)
20\( e - 1 \)
21\( 2 \)
22\( \frac{3}{2} - 1 \)
23\( \frac{1}{2} (1 + e^{-\pi}) \)
24\( \frac{\pi}{4} - \frac{1}{2} \)
25\( \frac{\sqrt{2} + \ln(\sqrt{2} + 1)}{2} \)
Topic 7: Volumes of Solids (Disk, Washer, and Shell Methods)
1\( \frac{\pi}{2} \)
2\( \frac{\pi}{3} \)
3\( 2\pi \)
4\( \frac{16\pi}{7} \)
5\( \frac{32\pi}{5} \)
6\( \frac{2\pi}{3} + \frac{\pi}{2} \)
7\( \frac{4\pi}{3} \)
8\( 8\pi \)
9\( \pi (\ln 4 - 1) \)
10\( \frac{\pi}{2} \)
11\( \frac{\pi}{12} \)
12\( \frac{\pi}{6} \)
13\( 2\pi \)
14\( \frac{\pi}{12} \)
15\( \frac{\pi}{6} \)
16\( \frac{8\pi}{3} \)
17\( \frac{\pi}{30} \)
18\( \pi (e^2 - 1) \)
19\( \frac{4\pi}{15} \)
20\( \pi \)
21\( \pi (e^2 + e^{-2} - 2) \)
22\( \frac{\pi^2}{16} \)
23\( \frac{7\pi}{4} \)
24\( \frac{\pi}{4} - \frac{\pi}{2} \)
25\( \frac{4\pi}{3} \)
Topic 8: Improper Integrals
1\( 1 \)
2\( 1 \)
3\( \frac{1}{4} \)
4\( \frac{1}{2} \)
5\( \frac{1}{3} \)
6\( \frac{1}{3} \)
7\( \frac{1}{3} \)
8\( 2 \)
9\( 2 \)
10\( \frac{1}{4} \)
11\( \frac{\pi}{2} \)
12\( 1 \)
13\( \frac{1}{2} \)
14\( \frac{\pi}{2} \)
15\( 1 \)
16\( \frac{1}{2} \)
17\( \frac{\pi}{4} \)
18\( 1 \)
19\( 2 \)
20\( \frac{2}{5} \)
21\( \frac{\pi}{4} \)
22\( 0 \)
23\( \pi \)
24\( \frac{\sqrt{\pi}}{2} e^{-1} \)
25\( \frac{\pi}{2} \)