Rules for Integration and Differentiation

A complete set of vocabulary flashcards covering basic rules for integration and differentiation from the RIT School of Mathematics and Statistics.

$$\int u^{n}\,du$$

$$=\frac{u^{n+1}}{n+1}+C$$

$$\int\frac{1}{u}\,du$$

$$=\ln|u|+C$$

$$\int e^{u}\,du$$

$$=e^{u}+C$$

$$\int a^{u}\,du$$

$$=\frac{a^u}{\ln(a)}+C$$

$$\int\sin(u)\,du$$

$$=-\cos(u)+C$$

$$\int\cos(u)\,du$$

$$=\sin(u)+C$$

$$\int\sec^2(u)\,du$$

$$=\tan(u)+C$$

$$\int\csc^2(u)\,du$$

$$=-\cot(u)+C$$

$$\int\sec(u)\tan(u)\,du$$

$$=\sec(u)+C$$

$$\int\csc(u)\cot(u)\,du$$

$$=-\csc(u)+C$$

$$\int\sec(u)\,du$$

$$=\ln|\sec(u)+\tan(u)|+C$$

$$\int\csc(u)\,du$$

$$=\ln|\csc(u)-\cot(u)|+C$$

$$\int\tan(u)\,du$$

$$=\ln|\sec(u)|+C$$

$$\int\cot(u)\,du$$

$$=\ln|\sin(u)|+C$$

$$\int\frac{1}{u^2+a^2}\,du$$

$$=\frac{1}{a}\tan^{-1}\left(\frac{u}{a}\right)+C$$

$$\int\frac{1}{\sqrt{a^2-u^2}}\,du$$

$$=\sin^{-1}\left(\frac{u}{a}\right)+C$$

$$\frac{d}{dx}[u^{n}]$$

$$=nu^{n-1}\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[u\cdot v]$$

$$=u\cdot\frac{dv}{dx}+v\cdot\frac{du}{dx}$$

$$\frac{d}{dx}\left(\frac{u}{v}\right)$$

$$=\frac{v \cdot\frac{du}{dx} - u \cdot\frac{dv}{dx}}{v^2}$$

$$\frac{d}{dx}[e^{u}]$$

$$\frac{d}{dx}[e^u] = e^u \cdot \frac{du}{dx}$$

$$\frac{d}{dx}[a^{u}]$$

$$=a^{u}\cdot\ln(a)\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\log_{a}(u)]$$

$$=\frac{1}{u \cdot\ln(a)}\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\ln(u)]$$

$$=\frac{1}{u}\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\sin^{-1}(u)]$$

$$=\frac{1}{\sqrt{1 - u^2}}\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\tan^{-1}(u)]$$

$$=\frac{1}{1 + u^2}\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\sin(u)]$$

$$=\cos(u)\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\cos(u)]$$

$$=-\sin(u)\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\tan(u)]$$

$$=\sec^2(u)\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\csc(u)]$$

$$=-\csc(u)\cot(u)\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\sec(u)]$$

$$=\sec(u)\tan(u)\cdot\frac{du}{dx}$$

$$\frac{d}{dx}[\cot(x)]$$

$$=-\csc^2(u)\cdot\frac{du}{dx}$$