Tablice wzorów przydatnych przy rozwiązywania całek i pochodnych.
Tabela Pochodnych #
| n | Pochodna | Wartość |
|---|---|---|
| 1 | $$f(x) = C$$ | $$f’(x) = 0$$ |
| 2 | $$f(x) = x^n$$ | $$f’(x)= n \cdot x^{n-1}$$ |
| 3 | $$f(x) = x$$ | $$f’(x) = 1$$ |
| 4 | $$f(x) = \frac{1}{x}$$ | $$f’(x) = \frac{-1}{x^2}$$ |
| 5 | $$f(x) = \sqrt{x}$$ | $$f’(x) = \frac{1}{2\sqrt{x}}$$ |
| 6 | $$f(x) = a^x$$ | $$f’(x) = a^x\ln a$$ |
| 7 | $$f(x) = e^x$$ | $$f’(x) = e^x$$ |
| 8 | $$f(x) =\ln x$$ | $$f’(x) = \frac{1}{x}$$ |
| 9 | $$f(x) =\log a^x$$ | $$f’(x) = \frac{log_ae}{x}$$ |
| 10 | $$f(x) = \sin x$$ | $$f’(x) = \cos{x}$$ |
| 11 | $$f(x) = \cos x$$ | $$f’(x) = -\sin{x}$$ |
| 12 | $$f(x) = \tan{x}$$ | $$f’(x) = \frac{1}{\cos^2 x}$$ |
| 13 | $$f(x) = \cot x$$ | $$f’(x) = \frac{1}{\sin^2 x}$$ |
| 14 | $$f(x) = \arcsin x$$ | $$f’(x) = \frac{1}{\sqrt{1-x^2}}$$ |
| 15 | $$f(x) = \arccos x$$ | $$f’(x) = \frac{-1}{\sqrt{1-x^2}}$$ |
| 16 | $$f(x) = \text{arccot} x$$ | $$f’(x) = \frac{-1}{1+x^2}$$ |
| 17 | $$f(x) = (u+v)’$$ | $$f’(x) = u’+v’$$ |
| 18 | $$f(x) = (c \cdot f(x))’$$ | $$f’(x) = c \cdot f’(x)$$ |
| 19 | $$f(x) = (u \cdot v)’$$ | $$f’(x) = u’v+uv’$$ |
| 20 | $$f(x) = (\frac{u}{v})’$$ | $$f’(x) = \frac{u’v-uv’}{v^2}$$ |
| 21 | $$[f(g(x)]’$$ | $$f’(x) = f’(g(x))$$ |
Wzory na pochodne #
$$(c \cdot f(x))’=c \cdot f’(x)$$ $$ (f(x)\pm g(x))’=f’(x) \pm g’(x)$$ $$(f(x)\cdot g(x))’=f’(x)+ g’(x)$$ $$(\frac{f(x)}{g(x)})’= \frac{f(x)’\cdot f(x)-g(x)’ \cdot f(x)}{(g(x))^2} $$ $$(\sqrt{x})’=(x^{\frac{1}{2}})’= \frac{1}{2}\cdot x^{-\frac{1}{2}} $$
Tabela Całek #
| n | Całka | Wartość |
|---|---|---|
| 1 | $$\int dx$$ | $$x+C$$ |
| 2 | $$\int adx$$ | $$ax+C$$ |
| 3 | $$\int x^n dx$$ | $$\frac{1}{n+1}x^{n+1}$$ |
| 4 | $$\int \frac{dx}{x}$$ | $$\ln \mid x \mid + C$$ |
| 5 | $$\int a^xdx$$ | $$\frac{1}{\ln x}a^x+C$$ |
| 6 | $$\int e^xdx$$ | $$e^x+C$$ |
| 7 | $$\int \sqrt x dx$$ | $$\frac{2}{3} \sqrt{x^3} +C$$ |
| 8 | $$\frac{1}{\sqrt x}dx$$ | $$2 \sqrt x +C$$ |
| 9 | $$\int \frac{dx}{ax+b}$$ | $$\frac{1}{a} \ln{\mid ax+b \mid=}+C$$ |
| 10 | $$\int \sin x dx$$ | $$-\cos x+C$$ |
| 11 | $$\int \cos x dx$$ | $$\sin x +C$$ |
| 12 | $$\int \tan x dx$$ | $$-\ln{\mid \cos x \mid} +C$$ |
| 13 | $$\int \cot x dx$$ | $$\ln{\mid \sin x \mid} +C$$ |
| 14 | $$\int \frac{dx}{\cos^2 x}$$ | $$\tan x +C$$ |
| 15 | $$\int \frac{dx}{\sin^2 x}$$ | $$-\cot x+C$$ |
| 16 | $$\int \frac{dx}{x^2+a^2}$$ | $$\frac{1}{a}\arctan {\frac{x}{a}} +C$$ |
| 17 | $$\int \frac{dx}{\sqrt{a^2-x^2}}$$ | $$\arcsin \frac{x}{a}+C$$ |
| 18 | $$\int \frac{dx}{\sqrt{x^2-a^2}}$$ | $$\ln{\mid x+\sqrt{x^2-a^2} \mid}+C$$ |
| 19 | $$\int (ax+b)^ndx$$ | $$\frac{1}{a(n+1)} (ax+b)^{n+1}+C$$ |
| 20 | $$\int \frac{dx}{a^2-x^2}$$ | $$\frac{1}{2a} \ln{\mid \frac{a+x}{a-x}\mid}+C, a>0 \land \vert x \vert$$ |
| 21 | $$\frac{dx}{ \sqrt{x^2+a^2}}$$ | $$\ln{\vert \frac{x+\sqrt{}x^2+a^2}{a } \vert}+C$$ |
| 22 | $$\int \frac{f’(x)}{f(x)}$$ | $$\ln{\vert f’(x) \vert}+C$$ |
Wzory na całki #
$$ ax^2+bx+C = a[(x+\frac{b}{2a})^2-\frac{ \Delta }{4a^2}] $$ $$ \int f(x) \cdot g’(x)dx=f(x) \cdot g(x) - \int f’(x) \cdot g(x)dx $$
Podstawienie uniwersalne #
Dla \(\int F(\sin x, \cos x)dx\) #
$$t = \tan \frac{x}{2}$$
$$\sin x = \frac{2t}{1+t^2}$$
$$\cos x = \frac{1-t}{1+t^2}$$
$$dx = \frac{2dt}{1+t^2}$$
Dla \(\int F(\sin^2 x, \cos^2 x, \sin \cdot \cos)dx\) #
$$t=\tan x$$
$$\sin^2 x = \frac{t^2}{t^2+1}$$
$$\cos^2 x = \frac{1}{t^2+1}$$
$$\sin^2 x = \frac{t^2}{t^2+1}$$
$$\sin x \cdot \cos x = \frac{t}{t^2+1}$$
$$dx = \frac{dt}{t^2+1}$$