\[ \int k \, dx = kx + C \quad \text{(where k is constant)} \]

\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(where } n \neq -1\text{)} \]

\[ \int \frac{1}{x} \, dx = \ln|x| + C \]

\[ \int e^x \, dx = e^x + C \]

\[ \int a^x \, dx = \frac{a^x}{\ln a} + C \quad \text{(where } a > 0, a \neq 1\text{)} \]

\[ \int e^{ax} \, dx = \frac{e^{ax}}{a} + C \]

\[ \int \sin x \, dx = -\cos x + C \]

\[ \int \cos x \, dx = \sin x + C \]

\[ \int \tan x \, dx = \ln|\sec x| + C = -\ln|\cos x| + C \]

\[ \int \cot x \, dx = \ln|\sin x| + C \]

\[ \int \sec x \, dx = \ln|\sec x + \tan x| + C \]

\[ \int \csc x \, dx = \ln|\csc x - \cot x| + C \]

\[ \int \sec^2 x \, dx = \tan x + C \]

\[ \int \csc^2 x \, dx = -\cot x + C \]

\[ \int \sec x \tan x \, dx = \sec x + C \]

\[ \int \csc x \cot x \, dx = -\csc x + C \]

\[ \int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1}x + C = -\cos^{-1}x + C \]

\[ \int \frac{1}{1+x^2} \, dx = \tan^{-1}x + C = -\cot^{-1}x + C \]

\[ \int \frac{1}{x\sqrt{x^2-1}} \, dx = \sec^{-1}x + C = -\csc^{-1}x + C \]

\[ \int \frac{1}{\sqrt{a^2-x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C \]

\[ \int \frac{1}{a^2+x^2} \, dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C \]

\[ \int \frac{1}{x\sqrt{x^2-a^2}} \, dx = \frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right) + C \]

\[ \int \frac{1}{x^2-a^2} \, dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C \]

\[ \int \frac{1}{a^2-x^2} \, dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C \]

\[ \int \frac{1}{\sqrt{x^2+a^2}} \, dx = \ln\left|x + \sqrt{x^2+a^2}\right| + C \]

\[ \int \frac{1}{\sqrt{x^2-a^2}} \, dx = \ln\left|x + \sqrt{x^2-a^2}\right| + C \]

\[ \int \sqrt{x^2+a^2} \, dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln\left|x + \sqrt{x^2+a^2}\right| + C \]

\[ \int \sqrt{x^2-a^2} \, dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln\left|x + \sqrt{x^2-a^2}\right| + C \]

\[ \int \sqrt{a^2-x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C \]

\[ \int e^x[f(x) + f'(x)] \, dx = e^x f(x) + C \]

\[ \int \frac{f'(x)}{f(x)} \, dx = \ln|f(x)| + C \]

\[ \int [f(x)]^n f'(x) \, dx = \frac{[f(x)]^{n+1}}{n+1} + C \quad (n \neq -1) \]

\[ \int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C \]

\[ \int \frac{dx}{\sqrt{2ax-x^2}} = \sin^{-1}\left(\frac{x-a}{a}\right) + C \]

\[ \int \sqrt{2ax-x^2} \, dx = \frac{x-a}{2}\sqrt{2ax-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x-a}{a}\right) + C \]

\[ \int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + C \]

\[ \int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C \]

\[ \int \tan^2 x \, dx = \tan x - x + C \]

\[ \int \cot^2 x \, dx = -\cot x - x + C \]

\[ \int \sin^3 x \, dx = -\cos x + \frac{\cos^3 x}{3} + C \]

\[ \int \cos^3 x \, dx = \sin x - \frac{\sin^3 x}{3} + C \]

\[ \int u \, dv = uv - \int v \, du \]

ILATE Rule for choosing u:
I - Inverse Trigonometric Functions
L - Logarithmic Functions
A - Algebraic Functions
T - Trigonometric Functions
E - Exponential Functions

\[ \int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x) \]

Common substitutions: Let u = inner function, then du = derivative of inner function

\[ \int_a^b f(x) \, dx = F(b) - F(a) \quad \text{(Fundamental Theorem)} \]

\[ \int_a^b f(x) \, dx = -\int_b^a f(x) \, dx \]

\[ \int_a^a f(x) \, dx = 0 \]

\[ \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx \]

\[ \int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx \]

\[ \int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx \]

\[ \int_{-a}^a f(x) \, dx = \begin{cases} 2\int_0^a f(x) \, dx & \text{if } f(x) \text{ is even} \\ 0 & \text{if } f(x) \text{ is odd} \end{cases} \]

\[ \int_0^{2a} f(x) \, dx = \int_0^a f(x) \, dx + \int_0^a f(2a-x) \, dx \]

\[ \int \sinh x \, dx = \cosh x + C \]

\[ \int \cosh x \, dx = \sinh x + C \]

\[ \int \tanh x \, dx = \ln(\cosh x) + C \]

\[ \int \text{sech}^2 x \, dx = \tanh x + C \]

\[ \int \text{csch}^2 x \, dx = -\coth x + C \]

For rational functions:
\[ \int \frac{P(x)}{Q(x)} \, dx \]
where degree of P(x) < degree of Q(x)

\[ \int \frac{px + q}{(x-a)(x-b)} \, dx = A\ln|x-a| + B\ln|x-b| + C \]

\[ \int \frac{px + q}{(x-a)^2} \, dx = A\ln|x-a| + \frac{B}{x-a} + C \]

\[ \int \sin^n x \, dx = -\frac{1}{n}\sin^{n-1}x \cos x + \frac{n-1}{n}\int \sin^{n-2}x \, dx \]

\[ \int \cos^n x \, dx = \frac{1}{n}\cos^{n-1}x \sin x + \frac{n-1}{n}\int \cos^{n-2}x \, dx \]

\[ \int \tan^n x \, dx = \frac{\tan^{n-1}x}{n-1} - \int \tan^{n-2}x \, dx \]

\[ \int \sec^n x \, dx = \frac{\sec^{n-2}x \tan x}{n-1} + \frac{n-2}{n-1}\int \sec^{n-2}x \, dx \]

\[ \int e^{ax} \sin(bx) \, dx = \frac{e^{ax}}{a^2 + b^2}[a\sin(bx) - b\cos(bx)] + C \]

\[ \int e^{ax} \cos(bx) \, dx = \frac{e^{ax}}{a^2 + b^2}[a\cos(bx) + b\sin(bx)] + C \]

\[ \int x e^{ax} \, dx = \frac{e^{ax}}{a^2}(ax - 1) + C \]

\[ \int x^n e^{ax} \, dx = \frac{x^n e^{ax}}{a} - \frac{n}{a}\int x^{n-1} e^{ax} \, dx \]