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

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

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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