Fundamental Theorem of Calculus:

Part one:
$$\int_a^bf(x)\:dx=F(b)-F(a)$$

Where $F(x)$ is an antiderivative of $f(x)$

Part two:
$$\dfrac{d}{dx}\int^x_af(t)\:dt=f(x)$$

Basic Integration Rules:

Constant Rule: $\displaystyle{\int} 0 \:dx =C$

Constant Multiple Rule: $\displaystyle{\int} Cf(x) \:dx = C\displaystyle{\int}f(x) \:dx$

Sum and Difference Rule: $\displaystyle{\int}\left[ f(x)\pm g(x) \right]\:dx = \displaystyle{\int}f(x) \:dx\pm \displaystyle{\int}g(x)\:dx$
Power Rule: $\displaystyle{\int}x^n \:dx= \dfrac{x^{n+1}}{n+1}+C$

Inverse Bounds Rule: $\displaystyle{\int^b_a}f(x)\:dx=-\displaystyle{\int^a_b}f(x)\:dx$
Interval Addition Rule: $\displaystyle{\int^b_a}f(x)\:dx+\displaystyle{\int_b^c}f(x)\:dx=\displaystyle{\int^c_a}f(x)\:dx$

Equal Limits Rule: $\displaystyle{\int^a_a}f(x)\:dx=0$

Basic Integration Techniques:

Integration by Parts:

$$\int u(x)v'(x)\:dx=u(x)v(x)-\int v(x)v'(x)\:dx$$

Integration by Substitution:

$$\int f(g(x))g'(x)\:dx= \int f(u)\:du$$

Where $u=g(x)$ and $du = g'(x)\:dx$

Exponential and Logarithmic Integrals

$$\begin{array}{lll} \displaystyle{\int} e^x \: dx = e^x + C && \displaystyle{\int} a^x \: dx = \dfrac{a^x}{\ln{a}} + C \\[10pt] \displaystyle{\int} \dfrac{1}{x} \: dx = \ln{|x|} + C && \displaystyle{\int} \dfrac{1}{x \ln{a}} \: dx = \log_a |x| + C \end{array}$$

Trigonometric Integrals

\[ \begin{align*} &\hspace{-2cm}\int \sin x \: dx = -\cos x + C &\int \sec x \tan x \: dx = \sec x + C \\ &\hspace{-2cm}\int \cos x \: dx = \sin x + C &\int \csc x \cot x \: dx = -\csc x + C \\ &\hspace{-2cm}\int \tan x \: dx = -\ln |\cos x| + C & \int \cot x \: dx = \ln |\sin x| + C\\ &\hspace{-2cm}\int \sec^2 x \: dx = \tan x + C & \int \csc^2 x \: dx = -\cot x + C\\ \end{align*} \]

Inverse Trigonometric Integrals

\[ \begin{align*} &\hspace{-0.5cm}\int \sin^{-1} x \: dx = x \sin^{-1} x + \sqrt{1-x^2} + C &\int \sec^{-1} x \: dx = x \sec^{-1} x – \ln |x + \sqrt{x^2 – 1}| + C \\ &\hspace{-.5cm}\int \cos^{-1} x \: dx = x \cos^{-1} x + \sqrt{1-x^2} + C &\int \csc^{-1} x \: dx = x \csc^{-1} x – \ln |x + \sqrt{x^2 – 1}| + C \\ &\hspace{-.5cm}\int \tan^{-1} x \: dx = x \tan^{-1} x – \dfrac{1}{2} \ln(x^2 + 1) + C & \int \cot^{-1} x \: dx = x \cot^{-1} x + \dfrac{1}{2} \ln(x^2 + 1) + C \end{align*} \]