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Қоидаи интегронии функсияҳо

\(1^o. \quad \int{Cf(x)}\,dx = C\int{f(x)}\,dx\)

\(2^o. \quad \int [f(x) \pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx\)

Формулаи Нютон - Лейбнитс.

Агар \(f(x)\) дар порчаи \([a, b]\) бефосила ва \(F(x)\) - функсияи ибтидоии он дар ин фосила бошад, он гоҳ баробарии зерин ҷой дорад:

\(\int \limits _{a}^{b}f(x)dx = {\Bigl .}F(x){\Bigl |}_{a}^{b} = F(b) - F(a)\)

Интегралҳои функсияҳои элементарӣ

\(1^o. \quad ~\int\!0\, dx = C\)

\(2^o. \quad ~\int\!a\,dx = ax +C\)

\(3^o. \quad ~\int\!x^n\,dx =  \begin{cases} \frac{x^{n+1}}{n+1} + C, & n \ne -1 \\ \ln \left|x \right| + C, & n=-1\end{cases}\)

\(4^o. \quad \int\!{dx \over {a^2+x^2}} = {1 \over a}\,\operatorname{arctg}\,\frac{x}{a} + C = - {1 \over a}\,\operatorname{arcctg}\,\frac{x}{a} + C\)

\(5^o. \quad \int\!{dx \over {x^2-a^2}} = {1 \over 2a}\ln \left|{x-a \over {x+a}}\right| + C\)

\(6^o. \quad \int\!\ln {x}\,dx = x \ln {x} - x + C\)

\(7^o. \quad \int \frac{dx}{x\ln x} = \ln|\ln x|+ C\)

\(8^o. \quad\int\!\log_b {x}\,dx = x\log_b {x} - x\log_b {e} + C = x\frac{\ln {x} - 1}{\ln b} + C\)

\(9^o. \quad \int\!e^x\,dx = e^x + C\)

\(10^o. \quad \int\!a^x\,dx = \frac{a^x}{\ln{a}} + C\)

\(11^o. \quad \int\!{dx \over \sqrt{a^2-x^2}} = \arcsin {x \over a} + C\)

\(12^o. \quad \int\!{-dx \over \sqrt{a^2-x^2}} = \arccos {x \over a} + C\)

\(13^o. \quad \int\!{dx \over x\sqrt{x^2-a^2}} = {1 \over a}\,\operatorname{arcsec}\,{|x| \over a} + C\)

\(14^o. \quad \int\!{dx \over \sqrt{x^2\pm a^2}} = \ln \left|{x + \sqrt {x^2\pm a^2}}\right| + C\)

\(15^o. \quad \int\!\sin{x}\, dx = -\cos{x} + C\)

\(16^o. \quad \int\!\cos{x}\, dx = \sin{x} + C\)

\(17^o. \quad \int\!\operatorname{tg}\, {x} \, dx = -\ln{\left| \cos {x} \right|} + C\)

\(18^o. \quad \int\!\operatorname{ctg}\, {x} \, dx = \ln{\left| \sin{x} \right|} + C\)

\(19^o. \quad \int\!\sec{x} \, dx = \ln{\left| \sec{x} + \operatorname{tg}\,{x}\right|} + C\)

\(20^o. \quad \int\!\csc{x} \, dx = -\ln{\left| \csc{x} + \operatorname{ctg}\,{x}\right|} + C\)

\(21^o. \quad \int\!\sec^2 x \, dx = \int\!{dx \over \cos^2 x} = \operatorname{tg}\,x + C\)

\(22^o. \quad \int\!\csc^2 x \, dx = \int\!{dx \over \sin^2 x} = -\operatorname{ctg}\,x + C\)

\(23^o. \quad \int\!\sec{x} \, \operatorname{tg}\,{x} \, dx = \sec{x} + C\)

\(24^o. \quad \int\!\csc{x} \, \operatorname{ctg}\,{x} \, dx = - \csc{x} + C\)

\(25^o. \quad \int\!\sin^2 x \, dx = \frac{1}{2}(x - \sin x \cos x) + C\)

\(26^o. \quad \int\!\cos^2 x \, dx = \frac{1}{2}(x + \sin x \cos x) + C\)

\(27^o. \quad \int\!\sin^n x \, dx = - \frac{\sin^{n-1} {x} \cos {x}}{n} + \frac{n-1}{n} \int\!\sin^{n-2}{x} \, dx, n\in\mathbb{N}, n\geqslant 2\)

\(28^o. \quad \int\!\cos^n x \, dx = \frac{\cos^{n-1} {x} \sin {x}}{n} + \frac{n-1}{n} \int\!\cos^{n-2}{x} \, dx, n\in\mathbb{N}, n\geqslant 2\)

\(29^o. \quad \int\!\operatorname{arctg}\,{x} \, dx = x \, \operatorname{arctg}\,{x} - \frac{1}{2}\ln{\left( 1 + x^2 \right)} + C\)