Derivatives & Antiderivatives (College Board AP® Calculus BC): Revision Note

Derivatives & antiderivatives

How can I find the indefinite integrals of common functions?

• Because differentiation and integration are inverse operations

  • you can 'reverse' what you know about derivatives to find indefinite integrals

• I.e. if $f^{\text{'}}\left(x\right)=g\left(x\right)$

  • then $\int g\left(x\right)dx=f\left(x\right)+C$

• This means that all your derivative results for common functions

  • have indefinite integral equivalents

Indefinite integrals of powers of x

• $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$ therefore

  • $\int x^{n}dx=\frac{1}{n+1}{x}^{n+1}+C,n\ne -1$

    • Note that you can't integrate $x^{-1}=\frac{1}{x}$ using this rule

      • The denominator in the fraction would become zero

      • $\frac{1}{x}$ must be integrated using logarithms

• Also note these two special cases

  • $\int kdx=kx+C$, where $k$ is a constant

  • $\int 0dx=C$

Indefinite integrals of exponentials and 1/x

• $\frac{d}{dx}\left(e^x\right)=e^x$ and $\frac{d}{dx}\left(e^{kx}\right)=k{e}^{kx}$ therefore

  • $\int e^{x}dx=e^x+C$

  • $\int e^{kx}dx=\frac{1}{k}{e}^{kx}+C$

• $\frac{d}{dx}\left(a^x\right)=a^x\ln a$ and $\frac{d}{dx}\left(a^{kx}\right)=a^{kx}k\ln a$ therefore

  • $\int a^{x}dx=\frac{1}{\ln a}{a}^x+C$

  • $\int a^{kx}dx=\frac{1}{k\ln a}{a}^{kx}+C$

• $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$ therefore

  • $\int \frac{1}{x}dx=\ln \left|x\right|+C$

    • Don't forget the modulus (absolute value) sign around the $x$

    • This allows the integral to be valid for negative values of $x$ as well as for positive values

• In the above formulas, $k$ is a real number constant and $a$ is a positive real number constant

Indefinite integrals of trigonometric functions

• $\frac{d}{dx}\left(\sin x\right)=\cos x$ and $\frac{d}{dx}\left(\sin kx\right)=k\cos kx$ therefore

  • $\int \cos xdx=\sin x+C$

  • $\int \cos kxdx=\frac{1}{k}\sin kx+C$

• $\frac{d}{dx}\left(\cos x\right)=-\sin x$ and $\frac{d}{dx}\left(\cos kx\right)=-k\sin kx$ therefore

  • $\int \sin xdx=-\cos x+C$

  • $\int \sin kxdx=-\frac{1}{k}\cos kx+C$

• $\frac{d}{dx}\left(\tan x\right)={\sec }^{2}x$ and $\frac{d}{dx}\left(\tan kx\right)=k{\sec }^{2}kx$ therefore

  • $\int {\sec }^{2}xdx=\tan x+C$

  • $\int {\sec }^{2}kxdx=\frac{1}{k}\tan kx+C$

• In the above formulas, $k$ is a real number constant

Indefinite integrals of reciprocal trigonometric functions

• $\frac{d}{dx}\left(\sec x\right)=\tan x\sec x$ and $\frac{d}{dx}\left(\sec kx\right)=k\tan kx\sec kx$ therefore

  • $\int \tan x\sec xdx=\sec x+C$

  • $\int \tan kx\sec kxdx=\frac{1}{k}\sec kx+C$

• $\frac{d}{dx}\left(\csc x\right)=-\cot x\csc x$ and $\frac{d}{dx}\left(\csc kx\right)=-k\cot kx\csc kx$ therefore

  • $\int \cot x\csc xdx=-\csc x+C$

  • $\int \cot kx\csc kxdx=-\frac{1}{k}\csc kx+C$

• $\frac{d}{dx}\left(\cot x\right)=-{\csc }^{2}x$ and $\frac{d}{dx}\left(\mathrm{co}tkx\right)=-k{\csc }^{2}kx$ therefore

  • $\int {\csc }^{2}xdx=-\cot x+C$

  • $\int {\csc }^{2}kxdx=-\frac{1}{k}\cot kx+C$

• In the above formulas, $k$ is a real number constant

Indefinite integrals using inverse trigonometric functions

• $\frac{d}{dx}\left(\arcsin x\right)=\frac{1}{\sqrt{1-x^2}},-1<x<1,$ therefore

  • $\int \frac{1}{\sqrt{1-x^2}}dx=\arcsin x+C,-1<x<1$

  • $\int \frac{1}{\sqrt{k^2-x^2}}dx=\arcsin \left(\frac{x}{k}\right)+C,-k<x<k$

• $\frac{d}{dx}\left(\mathrm{arc}\cos x\right)=-\frac{1}{\sqrt{1-x^2}},-1<x<1,$ therefore

  • $\int \frac{1}{\sqrt{1-x^2}}dx=-\mathrm{arc}\cos x+C,-1<x<1$

• You can see that $\int \frac{1}{\sqrt{1-x^2}}dx$ is either $\arcsin x+C$ or $-\arccos x+C$

  • Usually $\arcsin$ is used when finding indefinite integrals of this form

• $\frac{d}{dx}\left(\mathrm{arc}\tan x\right)=\frac{1}{1+x^2}$ therefore

  • $\int \frac{1}{1+x^2}dx=\mathrm{arc}\tan x+C$

  • $\int \frac{1}{k^2+x^2}dx=\frac{1}{k}\mathrm{arc}\tan \left(\frac{x}{k}\right)+C$

Table of common indefinite integrals

In the table below, $k$ is a real number constant and $a$ is a positive real number constant