Reference Materials for Mathematics

\begin{align*} \int_a^b f(x) dx \amp = - \int_b^a f(x) dx \\ \int_a^b cdx \amp = c(b-a) \\ \int_a^a f(x) dx \amp = 0 \\ \int_a^b f(x) dx + \int_b^c f(x) dx \amp = \int_a^c f(x) dx \\ f(x) \leq g(x) \amp \implies \int_a^b f(x) dx \leq \int_a^b g(x) dx \\ \int_a^b f(x) + g(x) dx \amp = \int_a^b f(x) dx + \int_a^b g(x) dx \\ \int_a^b f(x) - g(x) dx \amp = \int_a^b f(x) dx - \int_a^b g(x) dx \\ \int_a^b cf(x) dx \amp = c \int_a^b f(x) dx \end{align*}

\begin{align*} \int f(x) + g(x) dx \amp = \int f(x) dx + \int g(x) dx \\ \int f(x) - g(x) dx \amp = \int f(x) dx - \int g(x) dx \\ \int cf(x) dx \amp = c \int f(x) dx \end{align*}

\begin{align*} \frac{d}{dx} \int_a^x g(t) dt \amp = g(x) \\ \int_a^b \frac{d}{dx} f(x) dx \amp = f(b) - f(a) \\ \int_a^b f(x) dx \amp = F(b) - F(a) \text{ if } F \text{ satisfies } \frac{dF(x)}{dx} = f(x) \end{align*}

\begin{gather*} \int f(x) g^\prime(x) dx = f(x) g(x) - \int f^\prime(x) g(x) dx \end{gather*}

\begin{align*} \amp \text{Expression} \amp \amp \text{Substitution} \amp \amp \text{Identity} \amp \amp \text{Domain} \\ \amp \sqrt{a^2-x^2} \amp \amp x = a sin \theta \amp \amp \sqrt{a^2 - x^2} = a \cos \theta \amp \amp \frac{-\pi}{2} \leq \theta \leq \frac{\pi}{2} \\ \amp \sqrt{a^2+x^2} \amp \amp x = a \tan \theta \amp \amp \sqrt{a^2 + x^2} = a \sec \theta \amp \amp \frac{-\pi}{2} \lt \theta \lt \frac{\pi}{2} \\ \amp \sqrt{x^2-a^2} \amp \amp x = a \sec \theta \amp \amp \sqrt{x^2 - a^2} = a \tan \theta \amp \amp 0 \leq \theta \lt \frac{\pi}{2} \end{align*}

\begin{gather*} \int \frac{1}{x-a} dx = \ln |x-a| + c \\ \int \frac{2ax + b}{ax^2+bx + c} dx = \ln |ax^2 + bx + c| + c\\ \int \frac{1}{(x-a)^2 + b^2} dx = \frac{1}{b} \arctan \left( \frac{x-a}{b} \right) + c \end{gather*}

\begin{align*} \int x^ndx \amp = \frac{x^{n+1}}{n+1} + c \\ \int \frac{1}{x} dx \amp = \ln |x| + c \\ \int e^x dx \amp = e^x + c \\ \int a^x dx \amp = \frac{a^x}{\ln a}+ c \\ \int \ln x dx \amp = x \ln x - x + c \\ \int \log_a dx \amp = x \log_a x - \frac{x}{\ln a} + c \\ \int \sin x dx \amp = -\cos x + c \\ \int \cos x dx \amp = \sin x + c \\ \int \cos^2 x dx \amp = \frac{x}{2} + \frac{\sin 2x}{4} + c \\ \int \sin^2 x dx \amp = \frac{x}{2} - \frac{\sin 2x}{4} + c\\ \int \tan x dx \amp = \ln |\sec x| + c \\ \int \sec x dx \amp = \ln | \sec x + \tan x | + c \\ \int \csc x dx \amp = \ln | \csc x - \cot x | + c \\ \int \cot x dx \amp = \ln |\sin x| + c \\ \int \sec x \tan x dx \amp = \sec x + c \\ \int \csc x \cot x dx \amp = -\csc x + c \\ \int \sec^2 x dx \amp = \tan x + c \\ \int \csc^2 x dx \amp = -\cot x + c \\ \int \sinh x dx \amp = \cosh x + c \\ \int \cosh x dx \amp = \sinh x + c \\ \int \tanh x dx \amp = \ln \cosh x + c \\ \int \sech x dx \amp = \arctan | \sinh x| + c \\ \int \csch x dx \amp = \ln \left| \tanh \frac{1}{2} x \right| + c \\ \int \coth x dx \amp = \ln |\sinh x| + c \\ \int \sech x \tanh x dx \amp = - \sech x + c \\ \int \csch x \coth x dx \amp = -\csch x + c \\ \int \csch^2 x dx \amp = -\coth x + c \\ \int \sech^2 x dx \amp = \tanh x + c \\ \int \frac{1}{\sqrt{1-x^2}} dx \amp = \arcsin x + c \\ \int \frac{-1}{\sqrt{1-x^2}}dx \amp = \arccos x + c\\ \int \frac{1}{1+x^2} dx \amp = \arctan x + c \\ \int \frac{-1}{1+x^2} dx \amp = \text{arccot} x + c\\ \int \frac{1}{\sqrt{1+x^2}} dx \amp = \text{arcsinh} x + c \\ \int \frac{1}{\sqrt{x^2-1}} dx \amp = \text{arccosh} x + c\\ \int \frac{1}{1-x^2} dx \amp = \text{arctanh} x + c \\ \int \frac{1}{1-x^2} dx \amp = \text{arccoth} x + c\\ \int \sqrt{a^2 + x^2} dx \amp = \frac{1}{2} x \sqrt{a^2 + x^2} + \frac{a^2}{2} \ln \left( x + \sqrt{a^2 + x^2} \right) + c\\ \int \sqrt{x^2 - a^2} dx \amp = \frac{1}{2} x \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 \amp = \frac{1}{2} x \sqrt{a^2 - x^2} + \frac{a^2}{2} \arctan \left( \frac{x}{\sqrt{a^2 - x^2}} \right) + c \end{align*}