What are the Standard Integrals?

[Greg Bernhardt]

Definition/Summary

This article is a list of standard integrals, i.e. the integrals which are commonly used while evaluating problems and as such, are taken for granted. This is a reference article, and can be used to look up the various integrals which might help while solving problems.

Equations



Extended explanation

List of Standard Integrals

1. Integrals of Polynomial functions

i] \int x^n \,dx = \frac{x^{n + 1}}{n + 1} + C \hspace{0.25in} (n \ne -1)

ii] \int \frac{1}{x} \,dx = \log_e |x| + C

2. Integrals of Exponential functions

iii] \int e^x \,dx = e^x + C

iv] \int a^x \,dx = \frac{a^x}{\log_e a} + C

2. Integrals of Trignometric functions

v] \int \sin x \,dx = - \cos x + C

vi] \int \cos x \,dx = \sin x + C

vii] \int \sec^2 x \,dx = \tan x + C

viii] \int \csc^2 x \,dx = -\cot x + C

ix] \int \sec x \tan x \,dx = \sec x + C

x] \int \csc x \cot x \,dx = -\csc x + C

xi] \int \cot x \,dx = \log_e |\sin x| + C

xii] \int \tan x \,dx = -\log_e |\cos x| + C

xiii] \int \sec x \,dx = \log_e |\sec x + \tan x|\ +\ C\ = \cosh^{-1}(\sec x)\ +\ C

= sech^{-1}(\cos x)\ +\ C\ = \tanh^{-1}(\sin x)\ +\ C\ = \coth^{-1}(\csc x)\ +\ C​

xiv] \int \csc x \,dx = \log_e |\csc x - \cot x|\ +\ C\ = -\cosh^{-1}(\csc x)\ +\ C

= -sech^{-1}(\sin x)\ +\ C\ = -\tanh^{-1}(\cos x)\ +\ C\ = -\coth^{-1}(\sec x)\ +\ C​

]

3. Integrals of Hyperbolic Functions

xv] \int\sinh ax \,dx = \frac{1}{a}\cosh ax + C

xvi] \int\cosh ax \,dx = \frac{1}{a}\sinh ax + C

xvii] \int \tanh ax \,dx = \frac{1}{a}\log_e|\cosh ax| + C

xviii] \int \coth ax \,dx = \frac{1}{a}\log_e|\sinh ax| + C

xviii^A] \int sech x \,dx\ = \cos^{-1}(sech x)\ +\ C

= \sec^{-1}(\cosh x)\ +\ C\ = \tan^{-1}(\sinh x)\ +\ C\ = -\tan^{-1}(cosech x)\ +\ C
= \cot^{-1}(cosech x)\ +\ C\ = -\cot^{-1}(\sinh x)\ +\ C​

4. Integrals of Reciprocals of Quadratic and Root Quadratic functions

xix] \int \frac{1}{\sqrt{a^2 - x^2}} \,dx = \arcsin \left(\frac{x}{a}\right) + C

xx] \int - \frac{1}{\sqrt{a^2 - x^2}} \,dx = \arccos \left(\frac{x}{a}\right) + C

xxi] \int \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \arctan \left(\frac{x}{a}\right) + C

xxii] \int - \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \,\mathrm{arccot} \left(\frac{x}{a}\right) + C

xxiii] \int \frac{1}{x\sqrt{x^2 - a^2}} \,dx = \frac{1}{a} \,\mathrm{arcsec} \left(\frac{x}{a}\right)\ +\ C = \frac{1}{a} \arccos \left(\frac{a}{x}\right)\ +\ C

xxiv] \int - \frac{1}{x\sqrt{x^2 - a^2}} \,dx = \frac{1}{a} \,\mathrm{arccsc} \left(\frac{x}{a}\right)\ +\ C = \frac{1}{a} \arcsin \left(\frac{a}{x}\right)\ +\ C

xxv] \int \frac{1}{x^2 - a^2} \,dx = \frac{1}{2a} \log_e \left|\frac{x - a}{x + a}\right|\ +\ C = \frac{1}{a}\tanh^{-1} \left(\frac{a}{x}\right)\ +\ C

xxvi] \int \frac{1}{a^2 - x^2} \,dx = \frac{1}{2a} \log_e \left|\frac{a + x}{a - x}\right|\ +\ C = \frac{1}{a}\tanh^{-1} \left(\frac{x}{a}\right)\ +\ C

xxvii] \int \frac{1}{\sqrt{a^2 + x^2}} \,dx = \log_e |x + \sqrt{a^2 + x^2}|\ +\ C = \sinh^{-1} \left(\frac{x}{a}\right)\ +\ C

xxviii] \int \frac{1}{\sqrt{x^2 - a^2}} \,dx = \log_e |x + \sqrt{x^2 - a^2}|\ +\ C = \cosh^{-1} \left(\frac{x}{a}\right)\ +\ C

5. Integrals of Root Quadratic functions

xxix] \int \sqrt{a^2 - x^2} \,dx = \frac{x}{2} \sqrt{a^2 - x^2}\ +\ \frac{a^2}{2} \arcsin {\left(\frac{x}{a}\right)}\ +\ C

xxx] \int \sqrt{x^2 - a^2} \,dx = \frac{x}{2} \sqrt{x^2 - a^2}\ +\ \frac{a^2}{2} \log_e |x + \sqrt{x^2 - a^2}|\ +\ C

xxxi] \int \sqrt{x^2 + a^2} \,dx = \frac{x}{2} \sqrt{x^2 + a^2}\ +\ \frac{a^2}{2} \log_e |x + \sqrt{x^2 + a^2}|\ +\ C

6. Integrals of Inverse Trignometric Functions

xxxii] \int \arcsin x \,dx = x \arcsin x + \sqrt{1 - x^2} + C

xxxiii] \int \arctan x \,dx = x \arctan x - \frac{1}{2} \log_e |1 + x^2| + C

xxxiv] \int \mathrm{arcsec}\,x \,dx = x \,\mathrm{arcsec}\,x\ -\ \log_e |x + \sqrt{x^2 - 1}|\ +\ C

7. Definite Integrals

xxxv] \int_{-\infty}^{\infty}{e^{-x^2} \,dx} = \sqrt \pi

xxxvi] \int_0^{\infty} x^{n-1} e^{-x} \,dx = \Gamma(n)

xxxvii] \int_{-\infty}^{\infty}\frac{\sin x}{x} \,dx= \pi

xxxviii] \int_{-\infty}^{\infty}\frac{\sin^2{x}}{x^2} \,dx= \pi

* This entry is from our old Library feature. If you know who wrote it, please let us know so we can attribute a writer. Thanks!

 

[fresh_42]

See also (wrong language, but the formulas should speak for themselves):
https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_Integrale
https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_Bestimmte_Integrale