Skip to main content
  1. Posts/

Tablica całek i pochodnych

Patryk Kępa
Author
Patryk Kępa
Almost, barely, but basically a human being
Table of Contents

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 \(\int F(\sin^2 x, \cos^2 x, \sin \cdot \cos)dx\)
#

$$t=\tan x$$

$$\sin^2 x = \frac{t^2}{t^2+1}$$

$$\cos^2 x = \frac{1}{t^2+1}$$

$$\sin^2 x = \frac{t^2}{t^2+1}$$

$$\sin x \cdot \cos x = \frac{t}{t^2+1}$$

$$dx = \frac{dt}{t^2+1}$$