rag/knowledge_base/integration_basics.md

# Integration Basics

## Standard Integrals
- $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ for $n \neq -1$
- $\int \frac{1}{x} \, dx = \ln|x| + C$
- $\int e^x \, dx = e^x + C$
- $\int a^x \, dx = \frac{a^x}{\ln a} + C$
- $\int \sin x \, dx = -\cos x + C$
- $\int \cos x \, dx = \sin x + C$
- $\int \sec^2 x \, dx = \tan x + C$
- $\int \csc^2 x \, dx = -\cot x + C$
- $\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1} x + C$
- $\int \frac{1}{1+x^2} \, dx = \tan^{-1} x + C$

## Integration by Substitution
$\int f(g(x)) g'(x) \, dx = \int f(u) \, du$ where $u = g(x)$

## Integration by Parts
$\int u \, dv = uv - \int v \, du$

LIATE priority for choosing $u$: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential

## Partial Fractions
$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$
$\frac{P(x)}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2}$

## Definite Integrals Properties
- $\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$
- $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$
- $\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx$
- If $f$ is even: $\int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx$
- If $f$ is odd: $\int_{-a}^a f(x) \, dx = 0$