arc/notes
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

vault backup: 2025-03-20 11:41:15

Browse Source
This commit is contained in:
arc 2025-03-20 11:41:15 -06:00
parent 4976da28c2
commit 693a196db5
1 changed files with 18 additions and 16 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

34
education/math/MATH1210 (calc 1)/Integrals.md
View File

@ -2,7 +2,9 @@
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 $F'(x) = f(x)$
## Notation
The collection of all antiderivatives of a function $f$ is referred to as the *indefinite integral of $f$ with respect to $x$*, and is denoted by:
$$ \int f(x) dx $$
## Examples
> Find the antiderivative of the function $y = x^2$
@ -11,18 +13,18 @@ An antiderivative is useful when you know the rate of change, and you want to fi
## 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 $n \ne -1$ |
| $\dfrac{d}{dx} kx = k$ | $\int k \space dx = kx + C$ |
| $\dfrac{d}{dx} \ln \|x\| = \dfrac{1}{x}$ | <br>$\int \dfrac{1}{x}dx = \ln \|x\| + C$ |
| $\dfrac{d}{dx} e^x = e^x$ | <br>$\int e^x dx = e^x + C$ |
| $\dfrac{d}{dx} a^x = (\ln{a}) a^x$ | $\int a^xdx = \ln \|x\| + C$ |
| $\dfrac{d}{dx} \sin x = \cos x$ | $\int \cos(x) dx = \sin (x) + C$ |
| $\dfrac{d}{dx} \cos x = -\sin x$ | $\int \sin(x)dx = \sin x + C$ |
| $\dfrac{d}{dx} \tan{x} = \sec^2 x$ | $\int \sec^2(x)dx = \tan(x) + C$ |
| $\dfrac{d}{dx} \sec x = \sec x \tan x$ | $\int sec^2(x) dx = \sec(x) + C$ |
| $\dfrac{d}{dx} \sin^{-1} x = \dfrac{1}{\sqrt{1-x^2}}$ | $\int \sec(x) \tan(x) dx = \sec x + C$ |
| $\dfrac{d}{dx} \tan^{-1} x = \dfrac{1}{1+x^2}$ | \int \dfrac{1}{\sqrt{1}} |
| $\dfrac{d}{dx} k f(x) = k f'(x)$ | |
| $\dfrac{d}{dx} f(x) \pm g(x) = f'(x) \pm g'(x)$ | |
| 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 $n \ne -1$ |
| $\dfrac{d}{dx} kx = k$ | $\int k \space dx = kx + C$ |
| $\dfrac{d}{dx} \ln \|x\| = \dfrac{1}{x}$ | <br>$\int \dfrac{1}{x}dx = \ln \|x\| + C$ |
| $\dfrac{d}{dx} e^x = e^x$ | <br>$\int e^x dx = e^x + C$ |
| $\dfrac{d}{dx} a^x = (\ln{a}) a^x$ | $\int a^xdx = \ln \|x\| + C$ |
| $\dfrac{d}{dx} \sin x = \cos x$ | $\int \cos(x) dx = \sin (x) + C$ |
| $\dfrac{d}{dx} \cos x = -\sin x$ | $\int \sin(x)dx = \sin x + C$ |
| $\dfrac{d}{dx} \tan{x} = \sec^2 x$ | $\int \sec^2(x)dx = \tan(x) + C$ |
| $\dfrac{d}{dx} \sec x = \sec x \tan x$ | $\int sec^2(x) dx = \sec(x) + C$ |
| $\dfrac{d}{dx} \sin^{-1} x = \dfrac{1}{\sqrt{1-x^2}}$ | $\int \sec(x) \tan(x) dx = \sec x + C$ |
| $\dfrac{d}{dx} \tan^{-1} x = \dfrac{1}{1+x^2}$ | $\int \dfrac{1}{\sqrt{1+x^2}}dx = \tan^{-1}x + C$ |
| $\dfrac{d}{dx} k f(x) = k f'(x)$ | $\int k*f(x)dx = k\int f(x)dx$ |
| $\dfrac{d}{dx} f(x) \pm g(x) = f'(x) \pm g'(x)$ | $\int (f(x) \pm g(x))dx = \int f(x) dx \pm \int g(x) dx$ |