1.Answer: Integration (or Anti-differentiation).
2.Answer: Constant of integration.
3.Answer: $\frac{x^{n+1}}{n+1} + C$.
5.Answer: $\frac{x^6}{6} + C$.
6.Answer: $\ln|x| + C$ (or $\log|x| + C$).
7.Answer: $\frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$.
8.Answer: $\int x^{1/2} dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C$.
9.Answer: $\int x^{-1/2} dx = \frac{x^{1/2}}{1/2} + C = 2\sqrt{x} + C$.
10.Answer: $\frac{x^3}{3} + x^2 + x + C$.
11.Answer: $5 \left(\frac{x^5}{5}\right) + C = x^5 + C$.
12.Answer: $3x - \frac{4x^2}{2} + C = 3x - 2x^2 + C$.
13.Answer: $\frac{x^{5/2}}{5/2} + C = \frac{2}{5}x^{5/2} + C$.
15.Answer: $\frac{x^2}{2} + \ln|x| + C$.
16.Answer: $-\cos x + C$.
19.Answer: $-\cot x + C$.
21.Answer: $-\csc x + C$.
23.Answer: $\frac{a^x}{\ln a} + C$.
24.Answer: $\frac{3^x}{\ln 3} + C$.
25.Answer: $-\cos x + \sin x + C$.
26.Answer: $2e^x - 5\tan x + C$.
27.Answer: $\sin^{-1} x + C$ (or $-\cos^{-1} x + C$).
28.Answer: $\tan^{-1} x + C$.
29.Answer: $\sec^{-1} x + C$.
30.Answer: $\int (\sec^2 x - 1) dx = \tan x - x + C$.
31.Answer: Let $u = x^2, du = 2x dx$. $\int \cos(u) du = \sin(u) + C = \sin(x^2) + C$.
32.Answer: $\frac{e^{3x}}{3} + C$.
33.Answer: $-\frac{\cos(5x+2)}{5} + C$.
34.Answer: Let $u = x^2+1, du = 2x dx$. $\int \frac{du}{u} = \ln|u| + C = \ln|x^2+1| + C$.
35.Answer: Let $u = \ln x, du = \frac{1}{x} dx$. $\int u^2 du = \frac{u^3}{3} + C = \frac{(\ln x)^3}{3} + C$.
36.Answer: $\ln|\sec x| + C$ or $-\ln|\cos x| + C$.
37.Answer: $\ln|\sin x| + C$.
38.Answer: $\ln|\sec x + \tan x| + C$.
39.Answer: $\ln|\csc x - \cot x| + C$.
40.Answer: Let $u = \tan^{-1} x, du = \frac{1}{1+x^2} dx$. $\int e^u du = e^u + C = e^{\tan^{-1} x} + C$.
41.Answer: Let $u = e^x + 1, du = e^x dx$. $\int \frac{du}{u} = \ln|u| + C = \ln(e^x + 1) + C$.
42.Answer: Let $u = x^3, du = 3x^2 dx$. $\int e^u du = e^u + C = e^{x^3} + C$.
43.Answer: $\frac{\tan(2x)}{2} + C$.
44.Answer: $\frac{\ln|2x+3|}{2} + C$.
45.Answer: $\ln|f(x)| + C$.
46.Answer: $\frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$.
47.Answer: $\frac{1}{2a} \ln\left|\frac{x-a}{x+a}\right| + C$.
48.Answer: $\frac{1}{2a} \ln\left|\frac{a+x}{a-x}\right| + C$.
49.Answer: $\sin^{-1}\left(\frac{x}{a}\right) + C$.
50.Answer: $\ln|x + \sqrt{x^2 + a^2}| + C$.
51.Answer: $\ln|x + \sqrt{x^2 - a^2}| + C$.
52.Answer: $\frac{x}{2} \sqrt{a^2 - x^2} + \frac{a^2}{2} \sin^{-1}\left(\frac{x}{a}\right) + C$.
53.Answer: $\frac{A}{x-1} + \frac{B}{x-2}$.
54.Answer: $\frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x+2}$.
55.Answer: Here $a=2$. $\frac{1}{2}\tan^{-1}\left(\frac{x}{2}\right) + C$.
56.Answer: $u \int v dx - \int \left( \frac{du}{dx} \int v dx \right) dx$.
57.Answer: Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential.
58.Answer: $x$ (Algebraic comes before Trigonometric).
59.Answer: $\ln x$ (Logarithmic comes before Algebraic).
60.Answer: $u=x, v=e^x$. $x \int e^x dx - \int (1 \cdot e^x) dx = x e^x - e^x + C$.
61.Answer: $e^x f(x) + C$.
62.Answer: Here $f(x) = \sin x, f'(x) = \cos x$. Answer is $e^x \sin x + C$.
63.Answer: Here $f(x) = \frac{1}{x}, f'(x) = -\frac{1}{x^2}$. Answer is $e^x \left(\frac{1}{x}\right) + C = \frac{e^x}{x} + C$.
64.Answer: $u = \ln x, v = 1$. $\ln x \int 1 dx - \int (\frac{1}{x} \cdot x) dx = x \ln x - \int 1 dx = x \ln x - x + C$.
65.Answer: $f(x) = \tan x$ and $f'(x) = \sec^2 x$.
66.Answer: $F(b) - F(a)$.
68.Answer: $\left[ \frac{x^3}{3} \right]_0^1 = \frac{1^3}{3} - 0 = \frac{1}{3}$.
69.Answer: $[\sin x]_0^{\pi/2} = \sin(\pi/2) - \sin(0) = 1 - 0 = 1$.
70.Answer: Negative (or $-1$).
73.Answer: $2 \int_0^a f(x) dx$.
74.Answer: Odd function (since $\sin^3(-x) = (-\sin x)^3 = -\sin^3 x$).
75.Answer: Since $x^5$ is an odd function and the limits are $-1$ to $1$, the integral is $0$.
