Tato úloha neprošla kontrolou správnosti.

Přehled základních primitivních funkcí

Úloha číslo: 1257

  1. \(\int{0}dx=c;\,\, c \in R ,\,\, x\in R\)

  2. \(\int{x^b}dx=\frac{x^{b+1}}{b+1}+c;\,\, b,c \in R ,\,\, x \in R\)

  3. \(\int{\frac{1}{x}}dx=\ln{|x|}+c;\,\, c \in R;\,\, x \in R ,\,\, x\ne 0\)

  4. \(\int{e^x}dx={e^x}+c;\,\, c\in R ,\,\, x\in R\)

  5. \(\int{a^x}dx=\frac{a^{x}}{\ln{a}}+c;\,\, a,c, \in R; \,\, a>0;\,\, a \ne 1 ,\,\, x\in R\)

  6. \(\int{\sin{x}}dx=-\cos{x}+c;\,\, c \in R ,\,\, x\in R\)

  7. \(\int{\cos{x}}dx=\sin{x}+c;\,\, b, a,c \in R ,\,\, x\in R\)

  8. \(\int{\frac{1}{\sin^2{x}}}dx=\mathrm{cotan}\,{x}+c;\,\, c \in R;\,\, x\in \left(n \pi,(n+1)\pi\right)\,\, n \in Z\)

  9. \(\int{\frac{1}{\cos^2{x}}}dx=\tan{x}+c;\,\,,c \in R;\,\, x\in \left(n \frac{\pi}{2},(n+1)\frac{\pi}{2}\right)\,\, n \in Z\)

  10. \(\int{\frac{1}{1+x^2}}dx=\arctan{x}+c=\mathrm{arctg}\,{x}+b;\,\, b,c \in R,\,\, x\in R\)

  11. \(\int{\frac{1}{\sqrt{1-x^2}}}dx=\arcsin{x}+c=-a\arccos{x}+b;\,\, b, c \in R;\,\, x\in (-1{,}1)\)

  12. \(\int{\frac{1}{1-x^2}}dx=\begin{cases} \frac{1}{2}\ln{\frac{1+x}{1-x}}+c;\,\, |x| \ne 1 \\ \mathrm{arctg}\,h{x}+c;\,\, |x|<1 \\ \mathrm{arccotg}\,h{x}+c;\,\, |x|>1 \end{cases};\,\, c \in R\)

  13. \(\int{\sinh{x}}dx=\cosh{x}+c;\,\, c \in R,\,\, x\in R\)

  14. \(\int{\cosh{x}}dx=\sinh{x}+c;\,\, c \in R,\,\, x\in R\)

  15. \(\int{\frac{1}{\sinh^2{x}}}dx=\mathrm{cotan}\,h{x}+c;\,\, c \in R,\,\, x\in R, x\ne 0\)

  16. \(\int{\frac{1}{\cosh^2{x}}}dx=\tanh{x}+c;\,\, c \in R,\,\, x\in R\)

  17. \(\int{\frac{1}{\sqrt{1+x^2}}}dx=\mathrm{argsinh}\,{x}+c=\ln{(x+\sqrt{1+x^2})}+b;\,\, c,b \in R,\,\, x\in R\)

  18. \(\int{\frac{1}{\sqrt{x^2-1}}}dx=\begin{cases} \mathrm{argcosh}\,{x}+c;\,\, |x|<1 \\ \ln{(x+\sqrt{x^2-1})}+c;\,\, |x|>1 \end{cases};\,\, c \in R\)

Přičemž platí tyto dvě vlastnosti:

  1. \(\int{f(x)+g(x)}dx=\int{f(x)}dx+\int{g(x)}dx\)
  2. \(\int{kf(x)}dx=k\int{f(x)}dx;\,\, k \in R\)
    Úroveň náročnosti: Vysokoškolská úloha
    Úloha s vysvětlením teorie
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