A [[Function|function]] $F$ is *an* **antiderivative** of $f$ over an interval I if its [[Derivative|derivative]] is equal to $f$ for all $x$ in $I$ ($\forall x \in I: F^\prime(x) = f(x)$). Its most *general* form is $F(x) + C$, where $C$ is an arbitrary constant. - As constants become zero when differentiated, $C$ is added to $F$ to show that for *any* $C$, $(F(x) + C)^\prime = f(x)$. - [[Integration Strategy]] ## Strategies #### Common Antiderivatives | Function | *Particular* Antiderivative | | -------------------------- | --------------------------- | | $cf(x)$ | $cF(x)$ | | $f(x) + g(x)$ | $F(x) + G(x)$ | | $x^n (n \ne -1)$ | $\frac{x^{n+1}}{n+1}$ | | $\frac{1}{x}$ | $\ln{\|x\|}$ | | $e^x$ | $e^x$ | | $b^x$ | $\frac{b^x}{\ln{b}}$ | | $\cos{x}$ | $\sin{x}$ | | $\sin{x}$ | $-\cos{x}$ | | $\sec^2{x}$ | $\tan{x}$ | | $\sec{x}\tan{x}$ | $\sec{x}$ | | $\frac{1}{\sqrt{1 - x^2}}$ | $\arcsin{x}$ | | $\frac{1}{1 + x^2}$ | $\arctan{x}$ | | $\cosh{x}$ | $\sinh{x}$ | | $\sinh{x}$ | $\cosh{x}$ | ![[Substitution Rule]] #### Symmetric Functions Suppose that $f$ is [[Continuity|continuous]] over $[-a, a]$. - If $f$ is even ($f(-x) = f(x)$), then $\int_{-a}^{a}f(x)dx = 2\int_{0}^{a}f(x)dx$. - If $f$ is odd ($f(-x) = -f(x)$), then $\int_{-a}^{a}f(x)dx = 0$.