КР1 Шпаргалка

1. $\int dx=x+C$	11. $\int \frac{dx}{{\sin }^{2}x}=-\cot x+C$
2. $\int x^{\alpha }dx=\frac{x^{\alpha +1}}{\alpha +1},\alpha \ne -1$	12. $\int \frac{dx}{x^2+a^2}=\frac{1}{a}\arctan \frac{x}{a}$
3. $\int \frac{dx}{x}=\ln \Vert x\Vert +C$	13. $\int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin \frac{x}{a}$
4. $\int e^{x}dx=e^x+C$	14. $\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln \Vert \frac{x-a}{x+a}\Vert$
5. $\int a^{x}dx=\frac{a^x}{\ln a}+C$	15. $\int \frac{dx}{\sqrt{x^2\pm a^2}}=\ln \Vert x+\sqrt{x^2\pm a^2}\Vert$
6. $\int \sin xdx=-\cos x+C$	16. $\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\ln \Vert \frac{x+a}{x-a}\Vert$
7. $\int \cos xdx=\sin x+C$	17. $\int \sinh xdx=\cosh x+C$
8. $\int \frac{dx}{\sin x}=\ln \Vert \tan \frac{x}{2}\Vert +C$	18. $\int \cosh xdx=\sinh x+C$
9. $\int \frac{dx}{\cos x}=\ln \Vert \tan (\frac{x}{2}+\frac{\pi }{4})\Vert$	19. $\int \frac{dx}{{\cosh }^{2}x}=\tanh x+C$
10. $\int \frac{dx}{{\cos }^{2}x}=\tan x+C$	20. $\int \frac{dx}{{\sinh }^{2}x}=-\coth x+C$

Свойство $R(\sin x,\cos x)$	Замена	Дифференциал
$R(-\sin x,\cos x)=-R$	$t=\cos x$	$dt=-\sin xdx$
$R(\sin x,-\cos x)=-R$	$t=\sin x$	$dt=\cos xdx$
$R(-\sin x,-\cos x)=R$	$t=\tan x$	$dt=dx/{\cos }^{2}x$
Универсальная	$t=\tan \frac{x}{2}$	$\sin x=\frac{2t}{1+t^2},\cos x=\frac{1-t^2}{1+t^2},dx=\frac{2dt}{1+t^2}$

Корень	Подстановка	Преобразование
$\sqrt{l^2-y^2}$	$y=l\sin t$	$\sqrt{l^2-l^2{\sin }^{2}t}=l\cos t$
$\sqrt{l^2+y^2}$	$y=l\tan t$	$\sqrt{l^2+l^2{\tan }^{2}t}=l/\cos t$
$\sqrt{y^2-l^2}$	$y=l/\cos t$	$\sqrt{l^2/{\cos }^{2}t-l^2}=l\tan t$