Wolfram Research also operates another online service, the Wolfram Mathematica Online Integrator. Your email address will not be published. \(\left.\left|\int_{a}^{b} f(x) g(x) d x\right| \leq \sqrt{\left(\int_{a}^{b} f^{2}(x) d x\right)\left(\int_{a}^{b} g^{2}(x) d x\right)}\right)\) above inequality is known as schwarz – Buny ak ovsky inequality. With this Definite Integration Formulas list, you can learn definition, properties of definite Integrals, summation of series by intergration, and some other important formulas to solve complicated problems. For instance in. b.Integration formulas for Trigonometric Functions. (i) \(\int_{0}^{\pi / 2}\)sinnx dx = \(\int_{0}^{\pi / 2}\) cosnx dx = \(\frac{(n-1)}{n} \frac{(n-3)}{(n-2)} \ldots \frac{2}{3} \cdot 1\)(n is odd) For a list of definite integrals, see, Products of functions proportional to their second derivatives, Definite integrals lacking closed-form antiderivatives, Learn how and when to remove this template message, Supplément aux tables d'intégrales définies, List of integrals of irrational functions, List of integrals of trigonometric functions, List of integrals of inverse trigonometric functions, List of integrals of hyperbolic functions, List of integrals of inverse hyperbolic functions, List of integrals of exponential functions, List of integrals of logarithmic functions, Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович), Marichev, Oleg Igorevich (Маричев, Олег Игоревич), Integraltafeln oder Sammlung von Integralformeln, Integral Tables Or A Collection of Integral Formulae, A short table of integrals - revised edition, Victor Hugo Moll, The Integrals in Gradshteyn and Ryzhik, wxmaxima gui for Symbolic and numeric resolution of many mathematical problems, https://en.wikipedia.org/w/index.php?title=Lists_of_integrals&oldid=983365487, Short description is different from Wikidata, Articles lacking in-text citations from November 2013, Articles with unsourced statements from April 2013, Creative Commons Attribution-ShareAlike License, This article includes a mathematics-related. Go down deep enough into anything and you will find mathematics. If you want to contact me, probably have some question write me using the contact form or email me on Solve various math concepts calculations similar to Definite Integration during homework or assignments by taking help from the Onlinecalculator.guru provided Formulas List, Cheat Sheet, & Tables. Other useful resources include Abramowitz and Stegun and the Bateman Manuscript Project. There are some functions whose antiderivatives cannot be expressed in closed form. Applications of each formula can be found on the following pages. More compact collections can be found in e.g. (xvi) If f(x), g(x) are integrable on the interval (a, b) then A. Dieckmann, Table of Integrals (Elliptic Functions, Square Roots, Inverse Tangents and More Exotic Functions): This page was last edited on 13 October 2020, at 20:46. g. Integration by Parts. + 3) \[\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + c} \], 4) \[\int {{{[f(x)]}^n}f'(x)dx = \frac{{{{[f(x)]}^{n + 1}}}}{{n + 1}}} + c\], 6) \[\int {\frac{{f'(x)}}{{f(x)}}dx = \ln |f(x)| + c} \], 7) \[\int {{a^x}dx = \frac{{{a^x}}}{{\ln a}} + c} \], 8) \[{\int a ^{f(x)}}dx = \frac{{{a^{f(x)}}}}{{\ln a}} + c\], 10) \[\int {{e^{f(x)}}dx = {e^{f(x)}} + c} \], 12) \[\int {[f(x) \pm g(x)]dx = \int {f(x)dx \pm \int {g(x)dx} } } \], 13) \[\int {f(x) \cdot g(x)dx = f(x)\left( {\int {g(x)dx} } \right) – \int \left[ {f'(x)\left( {\int {g(x)dx} } \right)} \right]dx} \], 14) \[\int {\ln xdx = x(\ln x – 1) + c} \], 17) \[\int {\tan xdx = \ln \sec x} + c\] or \[ – \ln \cos x + c\], 18) \[\int {\cot xdx = \ln \sin x + c} \], 19) \[\int {\sec xdx = \ln (\sec x + \tan x) + c} \] or \[\ln \tan \left( {\frac{x}{2} + \frac{\pi }{4}} \right) + c\], 20) \[\int {\csc xdx = \ln (\csc x – \cot x) + c} \] or \[\ln \tan \frac{x}{2} + c\], 21) \[\int {{{\sec }^2}xdx = \tan x + c} \], 22) \[\int {{{\csc }^2}xdx = – \cot x + c} \], 23) \[\int {\sec x\tan xdx = \sec x + c} \], 24) \[\int {\csc x\cot xdx = – \csc x + c} \], 27) \[\int {\tanh xdx = \ln \cosh x + c} \], 28) \[\int {\coth xdx = \ln \sinh x + c} \], 29) \[\int {\sec {\text{h}}xdx = {{\tan }^{ – 1}}(\sinh x) + c} \], 30) \[\int {\csc {\text{h}}xdx = – {{\coth }^{ – 1}}(\cosh x)} \], 31) \[\int {\sec {{\text{h}}^2}xdx = \tanh x + c} \], 32) \[\int {\csc {{\text{h}}^2}xdx = – \coth x + c} \], 33) \[\int {\sec {\text{h}}x\tanh xdx = – \sec {\text{h}}x + c} \], 34) \[\int {\csc {\text{h}}x\coth xdx = – \csc {\text{h}}x + c} \], 35) \[\int {\frac{1}{{\sqrt {{a^2} – {x^2}} }}dx = {{\sin }^{ – 1}}\frac{x}{a}} + c\] or \[{\cos ^{ – 1}}\frac{x}{a} + c\], 36) \[\int {\frac{1}{{\sqrt {{x^2} – {a^2}} }}dx = {{\cosh }^{ – 1}}\frac{x}{a}} + c\] or \[\ln (x + \sqrt {{x^2} – {a^2}} ) + c\], 37) \[\int {\frac{1}{{\sqrt {{x^2} + {a^2}} }}dx = {{\sinh }^{ – 1}}\frac{x}{a} + c} \] or \[\ln (x + \sqrt {{x^2} + {a^2}} ) + c\], 38) \[\int {\frac{1}{{{a^2} – {x^2}}}dx = \frac{1}{a}{{\tanh }^{ – 1}}\frac{x}{a} + c} \]  or  \[\frac{1}{{2a}}\ln \left( {\frac{{a + x}}{{a – x}}} \right) + c\], 39) \[\int {\frac{1}{{{x^2} – {a^2}}}dx = – \frac{1}{a}{{\coth }^{ – 1}}\frac{x}{a} + c} \] or \[\frac{1}{{2a}}\ln \left( {\frac{{x – a}}{{x + a}}} \right) + c\], 40) \[\int {\frac{1}{{{x^2} + {a^2}}}dx = \frac{1}{a}{{\tan }^{ – 1}}\frac{x}{a} + c} \], 41) \[\int {\frac{1}{{x\sqrt {{a^2} – {x^2}} }}dx = – \frac{1}{a}\sec {{\text{h}}^{ – 1}}\frac{x}{a} + c}\] or \[ – \frac{1}{a}\ln \left( {\frac{{a + \sqrt {{a^2} – {x^2}} }}{x}} \right) + c\], 42) \[\int {\frac{1}{{x\sqrt {{x^2} – {a^2}} }}dx = \frac{1}{a}{{\sec }^{ – 1}}\frac{x}{a} + c} \], 43) \[\int {\frac{1}{{x\sqrt {{x^2} + {a^2}} }}dx = – \frac{1}{a}\csc {{\text{h}}^{ – 1}}\frac{x}{a} + c} \] or \[- \frac{1}{a}\ln \left( {\frac{{a + \sqrt {{x^2} + {a^2}} }}{x}} \right) + c\], 44) \[\int {\sqrt {{a^2} – {x^2}} } dx = \frac{1}{2}x\sqrt {{a^2} – {x^2}} + \frac{{{a^2}}}{2}{\sin ^{ – 1}}\frac{x}{a} + c\], 45) \[\int {\sqrt {{x^2} – {a^2}} dx = \frac{1}{2}x\sqrt {{x^2} – {a^2}} – \frac{{{a^2}}}{2}{{\cosh }^{ – 1}}\frac{x}{a} + c} \] or \[\frac{1}{2}x\sqrt {{x^2} – {a^2}} – \frac{{{a^2}}}{2}\ln \left( {x + \sqrt {{x^2} – {a^2}} } \right) + c\], 46) \[\int {\sqrt {{x^2} + {a^2}} dx = \frac{1}{2}x\sqrt {{x^2} + {a^2}} + \frac{{{a^2}}}{2}{{\sinh }^{ – 1}}\frac{x}{a} + c}\] or \[\frac{1}{2}x\sqrt {{x^2} + {a^2}} + \frac{{{a^2}}}{2}\ln \left( {x + \sqrt {{x^2} + {a^2}} } \right) + c\], 47) \[\int {{e^{ax}}\sin (bx + c)dx = \frac{{{e^{ax}}}}{{{a^2} + {b^2}}}\left[ {a\sin (bx + c) – b\cos (bx + c)} \right]} \], 48) \[\int {{e^{ax}}\cos (bx + c)dx = \frac{{{e^{ax}}}}{{{a^2} + {b^2}}}\left[ {a\cos (bx + c) + b\sin (bx + c)} \right]} \], 49) \[\int {\sin mx\cos nxdx = – \frac{{\cos (m + n)x}}{{2(m + n)}}} – \frac{{\cos (m – n)x}}{{2(m – n)}} + c\], 50) \[\int {\sin mx\sin nxdx = – \frac{{\sin (m + n)x}}{{2(m + n)}}} + \frac{{\sin (m – n)x}}{{2(m – n)}} + c\], 51) \[\int {\cos mx\cos nxdx = \frac{{\sin (m + n)x}}{{2(m + n)}}} + \frac{{\sin (m – n)x}}{{2(m – n)}} + c\], 52) \[\int {{{\sin }^{ – 1}}xdx = x{{\sin }^{ – 1}}x + \sqrt {1 – {x^2}} + c} \], 53) \[\int {{{\cos }^{ – 1}}xdx = x{{\cos }^{ – 1}}x – \sqrt {1 – {x^2}} + c} \], 54) \[\int {{{\tan }^{ – 1}}xdx = x{{\tan }^{ – 1}}x – \frac{1}{2}\ln (1 + {x^2}) + c} \], 55) \[\int {{{\cot }^{ – 1}}xdx = x{{\cot }^{ – 1}}x + \frac{1}{2}\ln (1 + {x^2}) + c} \], 56) \[\int {{{\sec }^{ – 1}}xdx = x{{\sec }^{ – 1}}x – \ln \left( {x + \sqrt {{x^2} – 1} } \right) + c} \], 57) \[\int {{{\csc }^{ – 1}}xdx = x{{\csc }^{ – 1}}x + \ln \left( {x + \sqrt {{x^2} – 1} } \right) + c} \], 58) \[\int {\frac{1}{{a + b\sin x}}dx = \frac{2}{{\sqrt {{a^2} – {b^2}} }}{{\tan }^{ – 1}}\left( {\frac{{a{{\tan }^{ – 1}}\frac{x}{2} + b}}{{\sqrt {{a^2} – {b^2}} }}} \right) + c} \] if $${a^2} > {b^2}$$, 59) \[\int {\frac{1}{{a + b\sin x}}dx = \frac{1}{{\sqrt {{a^2} – {b^2}} }}\ln \left( {\frac{{a\tan \frac{x}{a} + b – \sqrt {{b^2} – {a^2}} }}{{a\tan \frac{x}{a} + b + \sqrt {{b^2} – {a^2}} }}} \right) + c} \] if $${a^2} < {b^2}$$, 60) \[\int {\frac{1}{{a + b\cos x}}dx = \frac{2}{{\sqrt {{a^2} – {b^2}} }}{{\tan }^{ – 1}}\left( {\sqrt {\frac{{a – b}}{{a + b}}} \tan \frac{x}{2}} \right) + c} \] if \[{a^2} > {b^2}\], 61) \[\int {\frac{1}{{a + b\cos x}}dx = \frac{1}{{\sqrt {{a^2} – {b^2}} }}\ln \left( {\frac{{\sqrt {b + a} + \tan \frac{x}{2}\sqrt {b – a} }}{{\sqrt {b + a} – \tan \frac{x}{2}\sqrt {b – a} }}} \right) + c}\] if $${a^2} < {b^2}$$, 62) \[\int {\frac{1}{{a + b\sinh x}}dx = \frac{1}{{\sqrt {{a^2} + {b^2}} }}\ln \left( {\frac{{\sqrt {{a^2} + {b^2}} + a\tanh \frac{x}{2} – b}}{{\sqrt {{a^2} + {b^2}} – a\tanh \frac{x}{2} + b}}} \right) + c} \], 63) \[\int {\frac{1}{{a + b\cosh x}}dx = \frac{{\sqrt {a + b} + \sqrt {a – b} \tanh \frac{x}{2}}}{{\sqrt {a + b}- \sqrt {a – b} \tanh \frac{x}{2}}} + c} \] if $$a > b$$, 64) \[\int {\frac{1}{{a + b\cosh x}}dx = \frac{2}{{\sqrt {{b^2} – {a^2}} }}{{\tan }^{ – 1}}\sqrt {\frac{{b – a}}{{b + a}}} {{\tanh }^{ – 1}}\frac{x}{2} + c} \] if $$a < b$$.