The list

The big list

Explicit

This section takes us through explicit antiderivatives, and the forms we will attempt to convert each integral into.

Fundamentals

Integrals using only the definition and linearity. Let F^\prime(x) = f(x) and G^\prime(x) = g(x)

\int f(x) \, dx
\int f(x) + g(x) \, dx
\int k f(x) \, dx
\int -f(x) \, dx
\int af(x) + bg(x) \, dx

Inverse power rule

\int x \, dx
\int x^2 \, dx
\int 2x^2 - 3x - 1 \, dx
\int 2x(x^2 + 4)^2
\int \frac{1}{x^2} \, dx
\int \frac{3}{x^6} + \frac{7}{x^{10}} - \frac{4}{x^{12}} \, dx
\int \sqrt{x} \, dx
\int \frac{1}{\sqrt{x}} \, dx
\int \sqrt[3]{x} + 2\sqrt[5]{x^2}\, dw
\int x^\pi \, dx

Logarithms and Exponents

\int e^x \, dx
\int 5e^{5x} \, dx
\int e^{x + a} \, dx
\int \frac{1}{e^x} \, dx
\int a^x \, dx
\int a^{bx} \, dx
\int \frac{1}{x} \, dx
\int \frac{1}{x + a} \, dx

Trigonometric Functions

\int \sin x \, dx
\int \cos x \, dx
\int \frac{1}{\sin^2 x} \, dx
\int \frac{1}{\cos^2 x} \, dx
\int \frac{\sin x}{\cos^2 x} \, dx
\int \frac{\cos x}{\sin^2 x} \, dx
\int \frac{1}{x^2 + 1} \, dx
\int \frac{1}{\sqrt{1 - x^2}} \, dx
\int \frac{1}{x\sqrt{x^2 - 1}} \, dx
\int \sinh x \, dx
\int \cosh x \, dx
\int \frac{1}{\sinh^2 x} \, dx
\int \frac{1}{\cosh^2 x} \, dx
\int \frac{\sinh x}{\cosh^2 x} \, dx
\int \frac{\cosh x}{\sinh^2 x} \, dx
\int \frac{1}{1 - x^2} \, dx
\int \frac{1}{\sqrt{x^2 + 1}} \, dx
\int \frac{1}{x\sqrt{1 - x^2}} \, dx

General Methods

Now we use some methods to expand the number of integrals we can solve.

Substitution

\int 2x(x^2 + 4)^2 \, dx
\int \frac{x}{\sqrt{1-2x^2}} \, dx

Integration by parts

\int \ln x \, dx
\int x\ln x \, dx
\int x^2 \ln x \, dx
\int x^n \ln x \, dx
\int \arctan x \, dx
\int e^x \cos x \, dx
\int e^x \sin x \, dx

Partial Fraction Decomposition

\int \frac{1}{(x-1)(x-2)} \, dx
\int \frac{1}{(x+a)(x+b)} \, dx
\int \frac{x-9}{(x+5)(x-2)} \, dx
\int \frac{x+c}{(x+a)(x+b)} \, dx
\int \frac{x^2}{x^2 - 1} \, dx
\int \frac{x^3 + x^2 + 2x + 1}{(x^2 + 1)(x^2 + 2)} \, dx

Specialized Methods

Trigonometric Identities

Trigonometric Substitution

Completing the square

Weirstrass Substitution

https://math.stackexchange.com/questions/967117/integration-of-int-frac1-x1x-frac13