Интеграл табыу

Интеграл табыу — математика анализда дифференцил табыу менән нигеҙ операцияһы. 

## Ингтеграл табыу ҡағиҙәләре

:\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

### [[Рациональ функциялар]]
: ~\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
 = \frac {1} {a^2+x^2}
}}
: \int\!{dx \over {x^2-a^2}} = {1 \over 2a}\ln \left|{x-a \over {x+a}}\right| + C

### [[Логарифмдар]]
: \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]]
: \int\!e^x\,dx = e^x + C
: \int\!a^x\,dx = \frac{a^x}{\ln{a}} + C

### [[Иррациональ функциялар]]
: \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}\, + 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]]
: \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

 =
\frac{1}{\operatorname{ch}\,x} = \operatorname{sech}\,x.

Иҫбатлау \int \operatorname{sech}\,x \, dx = 2\, \operatorname{arctg}\,\left(\operatorname{th}\, \frac{x}{2}\right) + C тикшерәбеҙ:
: \left(2\, \operatorname{arctg}\,\left(\operatorname{th}\, \frac{x}{2}\right) + C\right)' =
\frac{\operatorname{sech}^2\frac{x}{2}}{\operatorname{th}^2\frac{x}{2} + 1} =
\frac{1}{\operatorname{sh}^2\frac{x}{2}+\operatorname{ch}^2\frac{x}{2}} =
\frac{1}{\operatorname{ch}\,x} = \operatorname{sech}\,x.
}}