|
<center>
<table cellspacing=5>
{| cellspacing="5"
<tr>
|-
<td><math>\int x^n\,dx</math>
<td><math>=\frac{x^{n+1}}{n+1}+c</math>
<td><math>(\int x^n\in\mathbb{R},n\in\mathbb{N})dx</math>
<tr>
<td><math>=\int frac{x^\alpha\,dx{n+1}}{n+1}+c</math>
<td><math>=\frac{x^{\alpha+1}}{\alpha+1}+c</math>
<td><math>(x\in\mathbb{R}^+,-1\neq\alphan\in\mathbb{RN})</math>
|-
<tr>
<td><math>\int\frac{1}{x}\,dx</math>
<td><math>=\,\ln|int x|+c^\alpha\,dx</math>
<td><math>(0\neq x\in\mathbb{R})</math>
<td><math>=\frac{x^{\alpha+1}}{\alpha+1}+c</math>
<tr>
<td><math>\int e^x\,dx</math>
<td><math>=(x\,ein\mathbb{R}^x+c,-1\neq\alpha\in\mathbb{R})</math>
|-
<td><math>(x\in\mathbb{R})</math>
<tr>
<td><math>\int a^\frac{1}{x}\,dx</math>
<td><math>=\frac{a^x}{\ln a}+c</math>
<td><math>(x=\in\mathbb{R},1\neq a\in\mathbb{R}^ln|x|+)c</math>
<tr>
<td><math>(0\int\sinneq x\,dxin\mathbb{R})</math>
|-
<td><math>=-\cos x\,+c</math>
<td><math>(x\in\mathbb{R})</math>
<td><math>\int e^x\,dx</math>
<tr>
<td><math>\int\cos x\,dx</math>
<td><math>=\sin ,e^x\,+c</math>
<td><math>(x\in\mathbb{R})</math>
<td><math>(x\in\mathbb{R})</math>
<tr>
|-
<td><math>\int\frac{1}{\sin^2x}\,dx</math>
<td><math>=-\mathrm{ctg}\,x\,+c</math>
<td><math>(k\pi\neqint a^x\in\mathbb{R},k\in\mathbb{Z})dx</math>
<tr>
<td><math>\int=\frac{1a^x}{\cos^2xln a}\,dx+c</math>
<td><math>=\mathrm{tg}\,x\,+c</math>
<td><math>(\frac{k\pi}{2}\neq x\in\mathbb{R},k1\neq a\in\mathbb{ZR}^+)</math>
|-
<tr>
<td><math>\int\mathrm{sh}\, x\,dx</math>
<td><math>=\mathrm{ch}int\,sin x\,+cdx</math>
<td><math>(x\in\mathbb{R})</math>
<td><math>=-\cos x\,+c</math>
<tr>
<td><math>\int\mathrm{ch}\, x\,dx</math>
<td><math>=(x\mathrmin\mathbb{shR}\, x\,+c)</math>
|-
<td><math>(x\in\mathbb{R})</math>
<tr>
<td><math>\int\frac{1}{\mathrm{sh}^2x}cos x\,dx</math>
<td><math>=-\mathrm{cth}\, x\,+c</math>
<td><math>(0=\neqsin x\in\mathbb{R}),+c</math>
<tr>
<td><math>(x\intin\fracmathbb{1R}{\mathrm{ch}^2x}\,dx)</math>
|-
<td><math>=\mathrm{th}\, x\,+c</math>
<td><math>(x\in\mathbb{R})</math>
<td><math>\int\frac{1}{\sin^2x}\,dx</math>
<tr>
<td><math>\int\frac{1}{1+x^2}\,dx</math>
<td><math>=-\mathrm{arc\,tgctg}\, x\,+c</math>
<td><math>(x\in\mathbb{R})</math>
<td><math>(k\pi\neq x\in\mathbb{R},k\in\mathbb{Z})</math>
<tr>
|-
<td><math>\int\frac{1}{1-x^2}\,dx</math>
<td><math>=\frac{1}{2}\ln\left|\frac{x+1}{x-1}\right|+c</math>
<td><math>\int\frac{1}{\cos^2x}\,dx</math>
<td><math>=\left\{{\mathrm{ar\,th}\,x+c\quad(1>|x|\in\mathbb{R})\atop\mathrm{ar\,cth}\,x+c\quad(1<|x|\in\mathbb{R})}\right.</math>
<tr>
<td><math>=\int\fracmathrm{1tg}{\sqrt{1-,x^2}}\,dx+c</math>
<td><math>=\mathrm{arc\,sin} x\,+c</math>
<td><math>(1>|\frac{k\pi}{2}\neq x|\in\mathbb{R},k\in\mathbb{Z})</math>
|-
<tr>
<td><math>\int\frac{1}{\sqrt{1+x^2}}\,dx</math>
<td><math>=\int\mathrm{ar\,sh}\, x+c\,dx</math>
<td><math>(x\in\mathbb{R})</math>
<td><math>=\mathrm{ch}\,x\,+c</math>
<tr>
<td><math>\int\frac{1}{\sqrt{x^2-1}}\,dx</math>
<td><math>=(x\lnin\left|x+\sqrtmathbb{x^2-1R}\right|+c)</math>
|-
<td><math>=\left\{\;{\mathrm{ar\,ch}\,x+c\quad\quad(1<x\in\mathbb{R})\atop\!-\mathrm{ar\,ch}(-x)+c\quad(1>x\in\mathbb{R})}\right.</math>
</table>
<td><math>\int\mathrm{ch}\, x\,dx</math>
<td><math>=\mathrm{sh}\, x\,+c</math>
<td><math>(x\in\mathbb{R})</math>
|-
<td><math>\int\frac{1}{\mathrm{sh}^2x}\,dx</math>
<td><math>=-\mathrm{cth}\, x\,+c</math>
<td><math>(0\neq x\in\mathbb{R})</math>
|-
<td><math>\int\frac{1}{\mathrm{ch}^2x}\,dx</math>
<td><math>=\mathrm{th}\, x\,+c</math>
<td><math>(x\in\mathbb{R})</math>
|-
<td><math>\int\frac{1}{1+x^2}\,dx</math>
<td><math>=\mathrm{arc\,tg}\, x\,+c</math>
<td><math>(x\in\mathbb{R})</math>
|-
<td><math>\int\frac{1}{1-x^2}\,dx</math>
<td><math>=\frac{1}{2}\ln\left|\frac{x+1}{x-1}\right|+c</math>
<td><math>=\left\{{\mathrm{ar\,th}\,x+c\quad(1>|x|\in\mathbb{R})\atop\mathrm{ar\,cth}\,x+c\quad(1<|x|\in\mathbb{R})}\right.</math>
|-
<td><math>\int\frac{1}{\sqrt{1-x^2}}\,dx</math>
<td><math>=\mathrm{arc\,sin} x\,+c</math>
<td><math>(1>|x|\in\mathbb{R})</math>
|-
<td><math>\int\frac{1}{\sqrt{1+x^2}}\,dx</math>
<td><math>=\mathrm{ar\,sh}\,x+c</math>
<td><math>(x\in\mathbb{R})</math>
|-
<td><math>\int\frac{1}{\sqrt{x^2-1}}\,dx</math>
<td><math>=\ln\left|x+\sqrt{x^2-1}\right|+c</math>
<td><math>=\left\{\;{\mathrm{ar\,ch}\,x+c\quad\quad(1<x\in\mathbb{R})\atop\!-\mathrm{ar\,ch}(-x)+c\quad(1>x\in\mathbb{R})}\right.</math>
|}
</center>
'''Nevezetes alesetek:'''
<table cellspacing=10>
{| cellspacing="10"
<tr>
|-
<td><math>\int f(ax+b)\,dx</math>
<td><math>=\frac{F(ax+b)}{a}+C</math>
<td><math>\int f(ax+b)\,dx</math>
<td> (a ''lineáris belső függvény esete)''
<tr>
<td><math>=\int [gfrac{F(xax+b)]^\alpha\,g'(x)\,dx}{a}+C</math>
| (a ''lineáris belső függvény esete)''
<td><math>\ =\frac{[g(x)]^{\alpha+1}}{\alpha+1}+C</math>
|-
<td><math>(\alpha\neq-1)</math>
<tr>
<td><math>\int\frac{ [g'(x)}{]^\alpha\,g'(x)}\,dx</math>
<td><math>\ =\ln|g(x)|+C</math>
<td><math>\ =\frac{[g(x)]^{\alpha+1}}{\alpha+1}+C</math>
<td>
<tr>
<td><math>(\alpha\neq-1)</math>
<td colspan=3>Az utolsó példa konkrét alkalmazásaiként kapjuk, hogy
|-
<tr>
<th><math>\int\mathrm{tg}\;x\,dx</math>
<td><math>=-\int\frac{\cosg'(x)}{\cosg(x)}\,dx</math>
<td><math>\ =-\ln|\cos x|+C</math>
<td><math>\ =\ln|g(x)|+C</math>
<tr>
|
<td colspan=3> illetve
|-
<tr>
| colspan="3" | Az utolsó példa konkrét alkalmazásaiként kapjuk, hogy
<th><math>\int\mathrm{ctg}\;x\,dx</math>
|-
<td><math>=\int\frac{\sin'(x)}{\sin(x)}\,dx</math>
! <td><math>=\ int\ln|mathrm{tg}\sin ;x|+C \,dx</math>
</table>
<td><math>=-\int\frac{\cos'(x)}{\cos(x)}\,dx</math>
<td><math>\ =-\ln|\cos x|+C</math>
|-
| colspan="3" | illetve
|-
! <math>\int\mathrm{ctg}\;x\,dx</math>
<td><math>=\int\frac{\sin'(x)}{\sin(x)}\,dx</math>
<td><math>=\ \ln|\sin x|+C </math>
|}
== Speciális integrálási módszerek ==
| 