Factsheet: List of integrals

Summary

A list of common (and some uncommon) integrals of functions.

Throughout, \(a,k\) are real numbers and \(C\) is the constant of integration.

Antiderivatives of polynomial, rational, exponential, logarithmic functions

function	antiderivative w.r.t \(x\)	notes
\(a\)	\(ax + C\)
\(ax^n\)	\(\dfrac{ax^{n+1}}{n+1} + C\)	\(n \in\mathbb{R},\, n \neq -1\)
\(ax^{-1}\)	\(a\ln|x| + C\)
\(\dfrac{a}{bx + c}\)	\(\dfrac{a}{b}\ln|bx + c| + C\)	\(b,c\in \mathbb{R}\)
\(\dfrac{a}{(bx + c)^n}\)	\(\dfrac{a(bx + c)^{1-n}}{b(1-n)} + C\)	\(b,c\in \mathbb{R}\)
\(ae^{kx}\)	\(\dfrac{a}{k}e^{kx} + C\)
\(a\ln(kx)\)	\(ax\ln|kx| - ax + C\)

Antiderivatives of trigonometric functions

function	antiderivative w.r.t \(x\)
\(a\sin(kx)\)	\(-\dfrac{a}{k}\cos(kx) + C\)
\(a\cos(kx)\)	\(\dfrac{a}{k}\sin(kx) + C\)
\(a\tan(kx)\)	\(\dfrac{a}{k}\ln|\sec(kx)| + C\)
\(a\cot(kx)\)	\(\dfrac{a}{k}\ln|\sin(kx)| + C\)
\(a\sec(kx)\)	\(\dfrac{a}{k}\ln\left|\tan(kx) + \sec(kx)\right| + C\)
\(a\csc(kx)\)	\(\dfrac{a}{k}\left(\ln\left|\sin\left(\frac{kx}{2}\right)\right| -\ln\left|\cos\left(\frac{kx}{2}\right)\right|\right) + C\)

Antiderivatives of some hyperbolic functions

function	antiderivative w.r.t \(x\)
\(a\sinh(kx)\)	\(\dfrac{a}{k}\cosh(kx) + C\)
\(a\cosh(kx)\)	\(\dfrac{a}{k}\sinh(kx) + C\)
\(a\tanh(kx)\)	\(\dfrac{a}{k}\ln|\cosh(kx)| + C\)
\(a\coth(kx)\)	\(\dfrac{a}{k}\ln|\sinh(kx)| + C\)

Standard forms that integrate to inverse trigonometric/hyperbolic functions

function	antiderivative w.r.t \(x\)
\(\dfrac{a}{\sqrt{1-k^2x^2}}\)	\(\dfrac{a}{k}\sin^{-1}(kx) + C\)
\(-\dfrac{a}{\sqrt{1-k^2x^2}}\)	\(\dfrac{a}{k}\cos^{-1}(kx) + C\)
\(\dfrac{a}{1+k^2x^2}\)	\(\dfrac{a}{k}\tan^{-1}(kx) + C\)
\(\dfrac{a}{\sqrt{1+k^2x^2}}\)	\(\dfrac{a}{k}\sinh^{-1}(kx) + C\)
\(\dfrac{a}{\sqrt{k^2x^2 - 1}}\)	\(\dfrac{a}{k}\cosh^{-1}(kx) + C\)
\(\dfrac{a}{\sqrt{1-k^2x^2}}\)	\(\dfrac{a}{k}\tanh^{-1}(kx) + C\)

 

---

---

Further reading

For more about where these came from, please see Guide: Introduction to integration and [Proof sheet: Antiderivatives of other common functions].

Version history

v1.0: created in 08/25 by tdhc.

This work is licensed under CC BY-NC-SA 4.0.