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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] [tex]\int x^n \,dx = \frac{x^{n + 1}}{n + 1} + C \hspace{0.25in} (n \ne -1)[/tex]
ii] [tex]\int \frac{1}{x} \,dx = \log_e |x| + C[/tex]
2. Integrals of Exponential functions
iii] [tex]\int e^x \,dx = e^x + C[/tex]
iv] [tex]\int a^x \,dx = \frac{a^x}{\log_e a} + C[/tex]
2. Integrals of Trignometric functions
v] [tex]\int \sin x \,dx = - \cos x + C[/tex]
vi] [tex]\int \cos x \,dx = \sin x + C[/tex]
vii] [tex]\int \sec^2 x \,dx = \tan x + C[/tex]
viii] [tex]\int \csc^2 x \,dx = -\cot x + C[/tex]
ix] [tex]\int \sec x \tan x \,dx = \sec x + C[/tex]
x] [tex]\int \csc x \cot x \,dx = -\csc x + C[/tex]
xi] [tex]\int \cot x \,dx = \log_e |\sin x| + C[/tex]
xii] [tex]\int \tan x \,dx = -\log_e |\cos x| + C[/tex]
xiii] [tex]\int \sec x \,dx = \log_e |\sec x + \tan x|\ +\ C\ = \cosh^{-1}(\sec x)\ +\ C[/tex]
xiv] [tex]\int \csc x \,dx = \log_e |\csc x - \cot x|\ +\ C\ = -\cosh^{-1}(\csc x)\ +\ C[/tex]
3. Integrals of Hyperbolic Functions
xv] [tex]\int\sinh ax \,dx = \frac{1}{a}\cosh ax + C[/tex]
xvi] [tex]\int\cosh ax \,dx = \frac{1}{a}\sinh ax + C[/tex]
xvii] [tex]\int \tanh ax \,dx = \frac{1}{a}\log_e|\cosh ax| + C[/tex]
xviii] [tex]\int \coth ax \,dx = \frac{1}{a}\log_e|\sinh ax| + C[/tex]
xviiiA] [tex]\int sech x \,dx\ = \cos^{-1}(sech x)\ +\ C[/tex]
4. Integrals of Reciprocals of Quadratic and Root Quadratic functions
xix] [tex]\int \frac{1}{\sqrt{a^2 - x^2}} \,dx = \arcsin \left(\frac{x}{a}\right) + C[/tex]
xx] [tex]\int - \frac{1}{\sqrt{a^2 - x^2}} \,dx = \arccos \left(\frac{x}{a}\right) + C[/tex]
xxi] [tex]\int \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \arctan \left(\frac{x}{a}\right) + C[/tex]
xxii] [tex]\int - \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \,\mathrm{arccot} \left(\frac{x}{a}\right) + C[/tex]
xxiii] [tex]\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[/tex]
xxiv] [tex]\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[/tex]
xxv] [tex]\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[/tex]
xxvi] [tex]\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[/tex]
xxvii] [tex]\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[/tex]
xxviii] [tex]\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[/tex]
5. Integrals of Root Quadratic functions
xxix] [tex]\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[/tex]
xxx] [tex]\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[/tex]
xxxi] [tex]\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[/tex]
6. Integrals of Inverse Trignometric Functions
xxxii] [tex]\int \arcsin x \,dx = x \arcsin x + \sqrt{1 - x^2} + C[/tex]
xxxiii] [tex]\int \arctan x \,dx = x \arctan x - \frac{1}{2} \log_e |1 + x^2| + C[/tex]
xxxiv] [tex]\int \mathrm{arcsec}\,x \,dx = x \,\mathrm{arcsec}\,x\ -\ \log_e |x + \sqrt{x^2 - 1}|\ +\ C[/tex]
7. Definite Integrals
xxxv] [tex]\int_{-\infty}^{\infty}{e^{-x^2} \,dx} = \sqrt \pi[/tex]
xxxvi] [tex]\int_0^{\infty} x^{n-1} e^{-x} \,dx = \Gamma(n)[/tex]
xxxvii] [tex]\int_{-\infty}^{\infty}\frac{\sin x}{x} \,dx= \pi[/tex]
xxxviii] [tex]\int_{-\infty}^{\infty}\frac{\sin^2{x}}{x^2} \,dx= \pi[/tex]
* 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!
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] [tex]\int x^n \,dx = \frac{x^{n + 1}}{n + 1} + C \hspace{0.25in} (n \ne -1)[/tex]
ii] [tex]\int \frac{1}{x} \,dx = \log_e |x| + C[/tex]
2. Integrals of Exponential functions
iii] [tex]\int e^x \,dx = e^x + C[/tex]
iv] [tex]\int a^x \,dx = \frac{a^x}{\log_e a} + C[/tex]
2. Integrals of Trignometric functions
v] [tex]\int \sin x \,dx = - \cos x + C[/tex]
vi] [tex]\int \cos x \,dx = \sin x + C[/tex]
vii] [tex]\int \sec^2 x \,dx = \tan x + C[/tex]
viii] [tex]\int \csc^2 x \,dx = -\cot x + C[/tex]
ix] [tex]\int \sec x \tan x \,dx = \sec x + C[/tex]
x] [tex]\int \csc x \cot x \,dx = -\csc x + C[/tex]
xi] [tex]\int \cot x \,dx = \log_e |\sin x| + C[/tex]
xii] [tex]\int \tan x \,dx = -\log_e |\cos x| + C[/tex]
xiii] [tex]\int \sec x \,dx = \log_e |\sec x + \tan x|\ +\ C\ = \cosh^{-1}(\sec x)\ +\ C[/tex]
[tex]= sech^{-1}(\cos x)\ +\ C\ = \tanh^{-1}(\sin x)\ +\ C\ = \coth^{-1}(\csc x)\ +\ C[/tex]
xiv] [tex]\int \csc x \,dx = \log_e |\csc x - \cot x|\ +\ C\ = -\cosh^{-1}(\csc x)\ +\ C[/tex]
[tex]= -sech^{-1}(\sin x)\ +\ C\ = -\tanh^{-1}(\cos x)\ +\ C\ = -\coth^{-1}(\sec x)\ +\ C[/tex]
]3. Integrals of Hyperbolic Functions
xv] [tex]\int\sinh ax \,dx = \frac{1}{a}\cosh ax + C[/tex]
xvi] [tex]\int\cosh ax \,dx = \frac{1}{a}\sinh ax + C[/tex]
xvii] [tex]\int \tanh ax \,dx = \frac{1}{a}\log_e|\cosh ax| + C[/tex]
xviii] [tex]\int \coth ax \,dx = \frac{1}{a}\log_e|\sinh ax| + C[/tex]
xviiiA] [tex]\int sech x \,dx\ = \cos^{-1}(sech x)\ +\ C[/tex]
[tex]= \sec^{-1}(\cosh x)\ +\ C\ = \tan^{-1}(\sinh x)\ +\ C\ = -\tan^{-1}(cosech x)\ +\ C[/tex]
[tex]= \cot^{-1}(cosech x)\ +\ C\ = -\cot^{-1}(\sinh x)\ +\ C[/tex]
[tex]= \cot^{-1}(cosech x)\ +\ C\ = -\cot^{-1}(\sinh x)\ +\ C[/tex]
4. Integrals of Reciprocals of Quadratic and Root Quadratic functions
xix] [tex]\int \frac{1}{\sqrt{a^2 - x^2}} \,dx = \arcsin \left(\frac{x}{a}\right) + C[/tex]
xx] [tex]\int - \frac{1}{\sqrt{a^2 - x^2}} \,dx = \arccos \left(\frac{x}{a}\right) + C[/tex]
xxi] [tex]\int \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \arctan \left(\frac{x}{a}\right) + C[/tex]
xxii] [tex]\int - \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \,\mathrm{arccot} \left(\frac{x}{a}\right) + C[/tex]
xxiii] [tex]\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[/tex]
xxiv] [tex]\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[/tex]
xxv] [tex]\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[/tex]
xxvi] [tex]\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[/tex]
xxvii] [tex]\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[/tex]
xxviii] [tex]\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[/tex]
5. Integrals of Root Quadratic functions
xxix] [tex]\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[/tex]
xxx] [tex]\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[/tex]
xxxi] [tex]\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[/tex]
6. Integrals of Inverse Trignometric Functions
xxxii] [tex]\int \arcsin x \,dx = x \arcsin x + \sqrt{1 - x^2} + C[/tex]
xxxiii] [tex]\int \arctan x \,dx = x \arctan x - \frac{1}{2} \log_e |1 + x^2| + C[/tex]
xxxiv] [tex]\int \mathrm{arcsec}\,x \,dx = x \,\mathrm{arcsec}\,x\ -\ \log_e |x + \sqrt{x^2 - 1}|\ +\ C[/tex]
7. Definite Integrals
xxxv] [tex]\int_{-\infty}^{\infty}{e^{-x^2} \,dx} = \sqrt \pi[/tex]
xxxvi] [tex]\int_0^{\infty} x^{n-1} e^{-x} \,dx = \Gamma(n)[/tex]
xxxvii] [tex]\int_{-\infty}^{\infty}\frac{\sin x}{x} \,dx= \pi[/tex]
xxxviii] [tex]\int_{-\infty}^{\infty}\frac{\sin^2{x}}{x^2} \,dx= \pi[/tex]
* 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!