Интеграл табыу

Википедия — ирекле энциклопедия мәғлүмәте
Унда күсергә: төп йүнәлештәр, эҙләү

Интеграл табыуматематика анализда дифференцил табыу менән нигеҙ операцияһы.

Ингтеграл табыу ҡағиҙәләре[үҙгәртергә]

\int cf(x)\,dx = c\int f(x)\,dx
\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx
\int [f(x) - g(x)]\,dx = \int f(x)\,dx - \int g(x)\,dx
\int f(x)g(x)\,dx = f(x)\int g(x)\,dx - \int \left(d[f(x)]\int g(x)\,dx\right)\,dx
\int f(ax+b)\,dx = {1 \over a} F(ax+b)\,+C

Элементар фунциялар интегралдары[үҙгәртергә]

Рациональ функциялар[үҙгәртергә]

~\int\!0\, dx = C
~\int\!a\,dx = ax +C
~\int\!x^n\,dx =  \begin{cases} \frac{x^{n+1}}{n+1} + C, & n \ne -1 \\ \ln \left|x \right| + C, & n=-1\end{cases}
\int\!{dx \over {a^2+x^2}} = {1 \over a}\,\operatorname{arctg}\,\frac{x}{a} + C = - {1 \over a}\,\operatorname{arcctg}\,\frac{x}{a} + C
\int\!{dx \over {x^2-a^2}} = {1 \over 2a}\ln \left|{x-a \over {x+a}}\right| + C

Логарифмдар[үҙгәртергә]

\int\!\ln {x}\,dx = x \ln {x} - x + C
\int \frac{dx}{x\ln x} = \ln|\ln x|+ C
\int\!\log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C = x\frac{\ln {x} - 1}{\ln b} + C

Экспоненттар[үҙгәртергә]

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

Иррациональ функциялар[үҙгәртергә]

\int\!{dx \over \sqrt{a^2-x^2}} = \arcsin {x \over a} + C
\int\!{-dx \over \sqrt{a^2-x^2}} = \arccos {x \over a} + C
\int\!{dx \over x\sqrt{x^2-a^2}} = {1 \over a}\,\operatorname{arcsec}\,{|x| \over a} + C
\int\!{dx \over \sqrt{x^2\pm a^2}} = \ln \left|{x + \sqrt {x^2\pm a^2}}\right| + C

Тригонометрия функциялар[үҙгәртергә]

\int\!\sin{x}\, dx = -\cos{x} + C
\int\!\cos{x}\, dx = \sin{x} + C
\int\!\operatorname{tg}\, {x} \, dx = -\ln{\left| \cos {x} \right|} + C
\int\!\operatorname{ctg}\, {x} \, dx = \ln{\left| \sin{x} \right|} + C
\int\!\sec{x} \, dx = \ln{\left| \sec{x} + \operatorname{tg}\,{x}\right|} + C
\int\!\csc{x} \, dx = -\ln{\left| \csc{x} + \operatorname{ctg}\,{x}\right|} + C
\int\!\sec^2 x \, dx = \int\!{dx \over \cos^2 x} = \operatorname{tg}\,x + C
\int\!\csc^2 x \, dx = \int\!{dx \over \sin^2 x} = -\operatorname{ctg}\,x + C
\int\!\sec{x} \, \operatorname{tg}\,{x} \, dx = \sec{x} + C
\int\!\csc{x} \, \operatorname{ctg}\,{x} \, dx = - \csc{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\!\sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int\!\sin^{n-2}{x} \, dx, n\in\mathbb{N}, n\geqslant 2
\int\!\cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int\!\cos^{n-2}{x} \, dx, n\in\mathbb{N}, n\geqslant 2
\int\!\operatorname{arctg}\,{x} \, dx = x \, \operatorname{arctg}\,{x} - \frac{1}{2}\ln{\left( 1 + x^2 \right)} + C

Гиперболоид функциялар[үҙгәртергә]

\int \operatorname{sh}\,x \, dx = \operatorname{ch}\,x + C
\int \operatorname{ch}\,x \, dx = \operatorname{sh}\,x + C
\int \frac{dx}{\operatorname{ch}^2\,x} = \operatorname{th}\,x + C
\int \frac{dx}{\operatorname{sh}^2\,x} = - \operatorname{cth}\,x + C
\int \operatorname{th}\,x \, dx = \ln |\operatorname{ch}\,x| + C
\int \operatorname{csch}\,x \, dx = \ln\left| \operatorname{th}\,{x \over2}\right| + C
\int \operatorname{sech}\,x \, dx = \operatorname{arctg}\,(\operatorname{sh}\,x) + C
также \int \operatorname{sech}\,x \, dx = 2\, \operatorname{arctg}\, (e^x) + C
также \int \operatorname{sech}\,x \, dx = 2\, \operatorname{arctg} \, \left(\operatorname{th}\, \frac{x}{2}\right) + C
\int \operatorname{cth}\,x \, dx = \ln|\operatorname{sh}\,x| + C