notes/education/math/MATH1210 (calc 1)/Integrals.md

Antiderivatives

An antiderivative is useful when you know the rate of change, and you want to find a point from that rate of change

A function F is said to be an antiderivative of f if F'(x) = f(x)

Examples

Find the antiderivative of the function y = x^2

1. We know that f'(x) = 2x^1

Formulas

Differentiation Formula	Integration Formula
\dfrac{d}{dx} x^n = nx^{x-1}	\int x^n dx = \dfrac{1}{n+1}x^{n+1}+ C for n \ne -1
\dfrac{d}{dx} kx = k	\int k \space dx = kx + C
\dfrac{d}{dx} \ln \|x\| = \dfrac{1}{x}
\dfrac{d}{dx} e^x = e^x
\dfrac{d}{dx} a^x = (\ln{a}) a^x
\dfrac{d}{dx} \sin x = \cos x
\dfrac{d}{dx} \cos x = -\sin x
\dfrac{d}{dx} \tan{x} = \sec^2 x
\dfrac{d}{dx} \sec x = \sec x \tan x
\dfrac{d}{dx} \sin^-1 x = \dfrac{1}{\sqrt{1-x^2}}
$\dfrac{d}{dx} \tan^-1 x = \dfrac{}