Pochodne i całki

| | POCHODNA | | | wzory |
|FUNKCJA |FUNKCJI | | | |
| | | | | FUNKCJA | POCHODNA |
| | | | | |FUNKCJI |
| y=[pic] | y’=-[pic] | | | y’=[pic] |
| y= c = const | y’= 0 | | | y’=[f(x) + g(x)]’ = f ’(x) + g’(x) |
| y= ax + b | y’= a | | | y’=[f(x) - g(x)]’ = f ’(x) - g’(x) |
| y= ax2 + bx + c | y’= 2ax + b | | | y’=[f(x) g(x)]’ = f ’(x) g(x) + f(x) g’(x) |
| y= x n | y’= n x n – 1 , n(Q | | | y’=[a f(x)]’ = a f ’(x) |
| y= [pic] | y’= [pic] | | | y’= [pic] |
| y= [pic] = x [pic] | y’= [pic]=[pic] | | | |
| | | | |y’=[g(f(x))]’ = g’(f(x)) f ’(x) |
| y= sin x | y’= cos x | | | y= log a x | y’= [pic] |
| y= cos x | y’= -sin x | | | y’=[ln(x+[pic] |
| y= tg x | y’= [pic]=1 + tg2x | | |y’=[pic] | |
| | | | | |y’=[xx]’=xx(lnx +1) |
| | | | | | |
| y= ctg x | y’= [pic]=-(1+ctg2x) | | | PRZYKŁADY |
| | | | | |
| | | | | FUNKCJA | POCHODNA |
| | | | | y= ctg5 9x | y’=-5ctg4 9x [pic]9 |
| y= a x | y’= ax ln a | | | | |
| | | | | y= e 2x | y’= 2 e 2x |
| y= e x | y’= e x | | | y = x 2 + 100x19-[pic] | y’= 2x+1900x18 |
| y= ln(a+b) | y’= [pic] | | | y= x 203 | y’= 203 x 202 |
| y= ln x | y’= [pic] | | | y = sin 2x | y’= cos 2x 2 |
| y= arc sin x | y’= [pic] | | | y = cos 73x | y’= 7cos63x(-sin3x)3 |
| y= arc cos x | y’= - [pic] | | | y = tg 4x | y’= [pic]4 |
| y= arc tg x | y’=[pic] | | | y = ln(1+x2) | y’ = [pic] |
| y= arc ctg x | y’= - [pic] | | | y= [pic] | y’=[pic] |
| CAŁKA | FUNKCJA | | | CAŁKA | FUNKCJA |
|FUNKCJI f |PIERWOTNA F | | | | |
| | | | | FUNKCJI f | PIERWOTNA F |
| [pic] | [pic] | | | [pic]=[pic] |
| [pic]dx | ax +c | | | [pic]f(ax+b)dx = [pic] |
| [pic]dx | [pic]+c | | | Całkowanie przez części |
| | | | |[pic] |
| [pic]dx | [pic]+c | | | [pic]; a ( -1 |
| [pic]dx ; n (Q\{-1} | [pic] + c | | | [pic] , n (-1 |
| [pic]dx | [pic]+c | | | [pic] |
| [pic]dx =[pic]dx | ln(x(+c | | | [pic] , a ( 0 |
| [pic]dx | -[pic]+ c | | | [pic] | |
| | | | | |ln(x + [pic]+c |
| [pic]dx | e x + c | | | [pic]arc tgx dx | x arc tgx - [pic]ln(x2+1)+c |
| [pic]dx | arc tg x + c | | | [pic]dx | x(ln2x – 2lnx + 2) + c |
| [pic]dx | arc sin x + c | | |[pic]f[g(x)]g’(x)dx =[pic]f(t)dt . Całkowanie przez podstawianie , gdzie t|
| | | | |= g(x) i dt = g’(x)dx |
| | | | | PRZYKŁADY |
| [pic]dx | x(lnx – 1) + c | | | [pic] | [pic] + c | |
| [pic] dx ...