Tablični integrali

Integraljenje je jedna od dve osnovne operacije u matematičkoj analizi.

Neke klase integrala se rešavaju šablonski, prema formulama ili pravilima koja opisuju kako se rešava integral, ako smo u stanju da konkretan integral prepoznamo među sledećim primerima.

Na ovoj stranici se nalaze neki osnovni neodređeni integrali.

Pravila za integraljenje opštih funkcija

Homogenost:

$\int cf(x)\,dx=c\int f(x)\,dx$

Aditivnost:

$\int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$

Parcijalna integracija:

$\int f(x)g(x)\,dx=f(x)\int g(x)\,dx-\int \left(d[f(x)]\int g(x)\,dx\right)$

Integrali jednostavnih funkcija

Racionalne funkcije

više integrala na: Lista integrala racionalnih funkcija

$\int \,dx=x+C$

$\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ ako }}n\neq -1$

$\int {\frac {1}{x}}\,dx=\ln {\left|x\right|}+C$

$\int {\frac {1}{x^{2}+a^{2}}}\,dx={\frac {1}{a}}\,\operatorname {arctg} {\frac {x}{a}}\ +C$

Logaritamske funkcije

više integrala na: Lista integrala logaritamskih funkcija

$\int \ln {x}\,dx=x\ln {x}-x+C$

$\int \log _{b}{x}\,dx=x\log _{b}{x}-x\log _{b}{e}+C$

Eksponencijalne funkcije

više integrala na: Lista integrala eksponencijalnih funkcija

$\int e^{x}\,dx=e^{x}+C$

$\int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C$

Iracionalne funkcije

više integrala na: Lista integrala iracionalnih funkcija

$\int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx=\arcsin {\frac {x}{a}}\ +C$

$\int {x \over {\sqrt {x^{2}-1}}}\,dx={\mbox{arcsec}}\,{x}+C$

Trigonometrijske funkcije

više integrala na: Lista integrala trigonometrijskih funkcija

$\int \cos {x}\,dx=\sin {x}+C$

$\int \sin {x}\,dx=-\cos {x}+C$

$\int \operatorname {tg} {x}\,dx=-\ln {\left|\cos {x}\right|}+C_{1}=\ln {\left|\sec {x}\right|}+C_{2}$

$\int \csc {x}\,dx=\ln {\left|\csc {x}-\cot {x}\right|}+C$

$\int \sec {x}\,dx=\ln {\left|\sec {x}+\operatorname {tg} {x}\right|}+C$

$\int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C$

---

$\int \sec ^{2}x\,dx=\operatorname {tg} x+C$

$\int \csc ^{2}x\,dx=-\cot x+C$

$\int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C$

$\int \cos ^{2}x\,dx={\frac {1}{2}}(x+\sin x\cos x)+C$

---

$\int \csc {x}\,\cot {x}\,dx=-\csc {x}+C$

$\int \operatorname {arctg} {x}\,dx=x\,\operatorname {arctg} {x}-{\frac {1}{2}}\ln(1+x^{2})+C$

Hiperboličke funkcije

više integrala na: Lista integrala hiperboličkih funkcija

$\int \sinh x\,dx=\cosh x+C$

$\int \cosh x\,dx=\sinh x+C$

$\int \operatorname {tgh} x\,dx=\ln(\cosh x)+C$

$\int {\mbox{csch}}\,x\,dx=\ln \left|\operatorname {tgh} {x \over 2}\right|+C$

$\int {\mbox{sech}}\,x\,dx=\operatorname {arctg} (\sinh x)+C$

$\int \coth x\,dx=\ln |\sinh x|+C$

Nesvojstveni (nepravi) integrali

$\int _{0}^{\infty }{{\sqrt {x}}\,e^{-x}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$

$\int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$

$\int _{0}^{\infty }{{\frac {x}{e^{x}-1}}\,dx}={\frac {\pi ^{2}}{6}}$

$\int _{0}^{\infty }{{\frac {x^{3}}{e^{x}-1}}\,dx}={\frac {\pi ^{4}}{15}}$

$\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}$

$\int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx=\Gamma (z)$

Literatura

• Milton Abramowitz and Irene Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
• I.S. Gradshteyn (И. С. Градштейн), I.M. Ryzhik (И. М. Рыжик); Alan Jeffrey, Daniel Zwillinger, editors. Table of Integrals, Series, and Products, seventh edition. Academic Press. 2007. ISBN 978-0-12-373637-6. Errata. (Several previous editions as well.)
• A.P. Prudnikov (А. П. Прудников), Yu.A. Brychkov (Ю. А. Брычков), O.I. Marichev (О. И. Маричев). Integrals and Series. First edition (Russian), volume 1–5, Nauka, 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press. 1988–1992. ISBN 978-2-88124-097-3. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
• Yu.A. Brychkov (Ю. А. Брычков). Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas. Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press. 2008. ISBN 978-1-58488-956-4.
• Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st edition. Chapman & Hall/CRC Press. 2002. ISBN 978-1-58488-291-6. (Many earlier editions as well.)
• Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln]
• Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)