What are the Standard Integrals?

In summary, this article provides a comprehensive list of standard integrals commonly used in problem-solving. It includes integrals of polynomial, exponential, trigonometric, hyperbolic, and reciprocal functions, as well as definite integrals and integrals of inverse trigonometric functions. This article serves as a reference for quickly looking up the necessary integrals while solving problems.
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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]
[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]

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!
 
Mathematics news on Phys.org

What is a standard integrals list?

A standard integrals list is a comprehensive list of common mathematical integrals, along with their corresponding solutions and techniques for solving them.

Why is a standard integrals list useful?

A standard integrals list is useful for quickly solving integrals in a variety of mathematical problems, as it provides a reference for commonly used integrals and their solutions.

Where can I find a standard integrals list?

A standard integrals list can be found in various mathematics textbooks, online resources, and scientific calculators.

How can I use a standard integrals list effectively?

To use a standard integrals list effectively, it is important to understand the techniques and methods used to solve different types of integrals. Practice and familiarity with the list can also improve efficiency in solving integrals.

Are there any limitations to a standard integrals list?

Yes, a standard integrals list may not include every possible integral and may not cover more complex integrals that require advanced techniques or methods to solve.

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