POCHODNE:

$${\left( C \right)^{'} = 0\backslash n}{\left( x^{n} \right)^{'} = \text{nx}^{n - 1}}{\left( x \right)^{'} = 1\backslash n}{\left( \frac{a}{x} \right)^{'} = - \frac{a}{x^{2}}\backslash n}{\left( \sqrt{x} \right)^{'} = \frac{1}{2\sqrt{x}}\backslash n}{\left( a^{x} \right)^{'} = a^{x}\ln a\backslash n}{\left( e^{x} \right)^{'} = e^{x}\backslash n}{\left( \operatorname{}x \right)^{'} = \frac{1}{x\ln a}\backslash n}{\left( \ln x \right)^{'} = \frac{1}{x}\backslash n}{\left( \sin x \right)^{'} = \cos x\backslash n}{\left( \cos x \right)^{'} = - \sin x\backslash n}{\left( \text{tgx} \right)^{'} = \frac{1}{\cos^{2}x}\backslash n}{\left( \text{ctgx} \right)^{'} = - \frac{1}{\cos^{2}x}\backslash n}{\left( \text{arc}\sin x \right)^{'} = \frac{1}{\sqrt{1 - x^{2}}}\backslash n}{\left( \text{arc}\cos x \right)^{'} = - \frac{1}{\sqrt{1 - x^{2}}}\backslash n}{\left( \text{arctgx} \right)^{'} = \frac{1}{x^{2} + 1}\backslash n}{\left( \text{arcctgx} \right)^{'} = - \frac{1}{x^{2} + 1}\backslash n}$$

$${\left\lbrack f\left( x \right) + g\left( x \right) \right\rbrack^{'} = f^{'}\left( x \right) + g^{'}\left( x \right)\backslash n}{\left\lbrack f\left( x \right) - g\left( x \right) \right\rbrack^{'} = f^{'}\left( x \right) - g^{'}\left( x \right)\backslash n}{\left\lbrack a*f\left( x \right) \right\rbrack^{'} = a*f^{'}\left( x \right)\backslash n}{\left\lbrack f\left( x \right)*g\left( x \right) \right\rbrack^{'} = f^{'}\left( x \right)g\left( x \right) + f\left( x \right)g\left( x \right)^{'}\backslash n}{\left\lbrack \frac{f\left( x \right)}{g\left( g \right)} \right\rbrack^{'} = \frac{f^{'\left( x \right)}g\left( x \right) - f\left( x \right)g^{'\left( x \right)}}{\left\lbrack g\left( x \right) \right\rbrack^{2}}\backslash n}$$

$${\sqrt[b]{x^{a}} = x^{\frac{a}{b}}\backslash n}{\frac{1}{x^{a}} = x^{- a}\backslash n}{a^{b} = e^{b\ln a}\backslash n}{cos2x = \cos^{2}x - \sin^{2}x}$$

CAŁKI:

$${\int_{}^{}\text{dx} = x + C\backslash n}{\int_{}^{}x^{n}dx = \frac{1}{n + 1}x^{n + 1} + C\backslash n}{\int_{}^{}\text{xdx} = \frac{1}{2}x^{2} + C\backslash n}{\int_{}^{}{\frac{1}{x}dx = \ln{\left| x \right| + C}}}{\int_{}^{}{a^{x}dx = \frac{a^{x}}{\ln a}} + C\backslash n}{\int_{}^{}{e^{x}dx = e^{x} + C}}$$

∫sinxdx = −cosx + C

∫cosxdx = sinx + C ∖ n

WŁASNOŚCI CAŁEK:

∫[f(x)+g(x)]dx = ∫f(x)dx + ∫g(x)dx ∖ n∫[f(x)−g(x)]dx = ∫f(x)dx − ∫g(x)dx ∖ n∫af(x)dx = a∫f(x)dx

METODY CAŁKOWANIA:

-bezpośrednie

-całkowanie przez podstawienie

-całkowanie przez części

-całki wymierne

-całki z pierwiastkami

-całki trygonometryczne