#### *Related to [[Derivative]] and [[Integration]]* --- ## Definition - The anti[[Derivative|derivative]] of a function f(x) is a function F(x) that's **derivative** is equal to f(x). - A function can have **multiple antiderivatives** so long as they all derive to the same **original equation**. >[!example] >$\text{\color{Bisque}\large Antiderivative example}$ >$\large \Big(3x^2\Big)\longrightarrow\Big(x^3\Big) \text{ or } \Big(x^3 + C\Big)$ - Antiderivatives are an incredibly important part of [[Integration]] since the process of integration requires you to find the **antiderivative of the integrand function.** **In fact most people will indicate that they are finding a function's antiderivative by writing it in the form of an *indefinite integral: $\int$***. --- ## Formatting/notation - It is typical for the antiderivative of **a function to be denoted using the same letter as its base function, but capital.** For example the antiderivative of $f(x)$ is usually written as $F(x)$ - The **integral symbol: $\int$ is used to denote that you are finding the antiderivative of the function proceeding the it.** >[!warning] >- Since the derivative of a constant is zero, its impossible to know if an equation's antiderivative had a constant term *(and what that term is)*. To represent this ambiguity we add the variable $C$ to the end of all antiderivative equations. > $\large\text{Dont forget to include}\color{LightCyan}+C\text{ !}$ --- ## Antiderivative "rules" >[!quote] >#### *Standard rules* >| ***Name*** | ***Rule*** | ***Example*** | >|-|-|-| >| **Power rule** | $\int x^n\rightarrow\color{LightCyan}\left(\frac{x^{n+1}}{n+1}+C\right)$ | $\int x^2\rightarrow\color{LightCyan}\left(\frac{x^3}{3}+C\right)$ >| **Constant multiple rule** | $\int kf(x)\rightarrow\color{LightCyan}\left( k\int f(x)\right)$ | $\int 4x^3\rightarrow\color{LightCyan}\left( 4\cdot\int x^3\right)$ >| **Sum rule** | $\int f(x)+ g(x)\rightarrow\color{LightCyan}\int f(x)+\int g(x)$ | $\int x^2+ 2x\rightarrow\color{LightCyan}\int x^2+\int 2x$ >| **Difference rule** | $\int f(x)-g(x)\rightarrow\color{LightCyan}\int f(x)-\int g(x)$ | $\int 3x-2x^2\rightarrow\color{LightCyan}\int 3x-\int 2x^2$ >| **Constant rule** | $\int k\rightarrow\color{LightCyan}\Big( kx+C\Big)$ | $\int 4\rightarrow\color{LightCyan}\Big( 4x+C\Big)$ >| **Natural log rule** | $\int\frac{1}{x}\rightarrow\color{LightCyan}\Big( \ln\vert x\vert +C\Big)$ | $\int\frac{1}{2x}\rightarrow\color{LightCyan}\left(\frac{\ln\vert 2x\vert}{2}+C\right)$ >| **Euler's rule** | $\int e^x\rightarrow\color{LightCyan}\Big( e^x+C\Big)$ | $\int e^{2x}\rightarrow\color{LightCyan}\left(\frac{e^{2x}}{2}+C\right)$ >| **Natural log rule** | $\begin{aligned}\int\ln(x)\rightarrow\color{LightCyan}\Big(x\ln(x)-x\Big)\\\tiny\color{grey}\text{done using integration by parts}\end{aligned}$ | $\text{\color{grey}no example}$ >*Note: The power rule **will not work** if $n$ is equal to -1* >[!quote] >#### *Standard trigonometric rules* > >| ***Name*** | ***Rule*** | >|-|-| >| **Cosine rule** | $\int \cos x\rightarrow\color{LightCyan}\Big(\sin x+C\Big)$ >| **Sine rule** | $\int \sin x\rightarrow\color{LightCyan}\Big(-\cos x+C\Big)$ >| **Tan rule** | $\int \tan(x)\rightarrow\color{LightCyan}\ln{\large\vert}\sec (x)\,{\large\vert}$ >| **Secant rule** | $\int\sec(x)\rightarrow\color{LightCyan}\ln{\large\vert}\sec(x)+\tan(x)\,{\large\vert}$ >| **Secant squared rule** | $\int \sec^2(x)\rightarrow\color{LightCyan}\Big( \tan(x)+C\Big)$ >| **Cosec rule** | $\int\csc(x)\longrightarrow\color{lightcyan}-\ln{\large\vert}\csc(x)+\cot(x){\large\vert}+C$ >| **Cosec squared rule** | $\int \csc^2(x)\rightarrow\color{LightCyan}\Big( -\cot(x)+C\Big)$ >| **Cotangent rule** | $\int\cot(x)\longrightarrow\color{lightcyan}\ln{\large\vert}\sin(x){\large\vert}+C$ >| **Cotangent squared rule** | $\int\cot^2(x)\longrightarrow\color{lightcyan}-\frac{1}{\tan(x)}-x+C\;\;{\color{grey}\small=}\;\;-\cot(x)-x+C$ >| **Tan squared rule** | $\int\tan^2(x)\rightarrow\color{LightCyan}\tan(x)-x+C$ >[!bug] >###### *Useful algebra tip:* >- To get rid of an absolute value you can replace it with a plus or minus sign --- ## Calculator >[!cite] > >--- ><iframe src="https://www.wolframalpha.com/calculators/integral-calculator/" width="850" height="1200"></iframe> >