A longer list of derivatives & integrals

(MathAcademy is having me review a lot of calc lately.)

$ \begin{array}{c|c|c} \text{Function / Rule } f(x) & \text{Derivative } \dfrac{d}{dx}[f(x)] & \text{Integral } \int f(x) \, dx \\ \hline c & 0 & cx + C \\ x & 1 & \dfrac{1}{2}x^2 + C \\ cx & c & \dfrac{1}{2}cx^2 + C \\ x^n & nx^{n-1} & \dfrac{x^{n+1}}{n+1} + C \quad (n \neq -1) \\ c f(x) & c f'(x) & c \int f(x) \, dx \\ f(x) \pm g(x) & f'(x) \pm g'(x) & \int f(x) \, dx \pm \int g(x) \, dx \\ f(x)g(x) & f'(x)g(x) + f(x)g'(x) & \text{Integration by Parts: } \int u \, dv = uv - \int v \, du \\ \dfrac{f(x)}{g(x)} & \dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2} & \text{n/a} \\ f(g(x)) & f'(g(x))g'(x) & \text{Substitution: } \int f(g(x))g'(x) \, dx = \int f(u) \, du \\ e^x & e^x & e^x + C \\ e^u & e^u u' & e^u + C \quad (\text{for } \int e^u \, du) \\ a^x & a^x \ln a & \dfrac{a^x}{\ln a} + C \quad (a > 0, a \neq 1) \\ a^u & a^u \ln(a) \cdot u' & \dfrac{a^u}{\ln a} + C \quad (\text{for } \int a^u \, du) \\ \ln x & \dfrac{1}{x} & x \ln x - x + C \\ \ln u & \dfrac{u'}{u} & \ln|u| + C \quad (\text{for } \int \dfrac{u'}{u} \, dx) \\ \log_a x & \dfrac{1}{x\ln a} & x \log_a x - \dfrac{x}{\ln a} + C \\ \sin x & \cos x & -\cos x + C \\ \cos x & -\sin x & \sin x + C \\ \tan x & \sec^2 x & \ln|\sec x| + C \\ \sec x & \sec x \tan x & \ln|\sec x + \tan x| + C \\ \csc x & -\csc x \cot x & -\ln|\csc x + \cot x| + C \\ \cot x & -\csc^2 x & \ln|\sin x| + C \\ \sin u & \cos u \cdot u' & -\cos u + C \quad (\text{for } \int \sin u \cdot u' \, dx) \\ \cos u & -\sin u \cdot u' & \sin u + C \quad (\text{for } \int \cos u \cdot u' \, dx) \\ \tan u & \sec^2 u \cdot u' & \ln|\sec u| + C \quad (\text{for } \int \tan u \cdot u' \, dx) \\ \arcsin x & \dfrac{1}{\sqrt{1-x^2}} & x \arcsin x + \sqrt{1-x^2} + C \\ \arccos x & -\dfrac{1}{\sqrt{1-x^2}} & x \arccos x - \sqrt{1-x^2} + C \\ \arctan x & \dfrac{1}{1+x^2} & x \arctan x - \dfrac{1}{2}\ln(1+x^2) + C \end{array} $