Jesteś tu: StudiaCałkiWzory całkowe wybranych funkcji

Wzory całkowe wybranych funkcji

Numer Wzór całkowy Przykład
\(\int dx=x+C\)
\(\int k\ dx=kx+C\) \(\int 7\ dx=7x+C\)
\(\int x^n\ dx=\frac{1}{n+1}x^{n+1}+C,\quad n\ne -1\) \(\int x^2\ dx=\frac{1}{3}x^{3}+C\)
\(\int \frac{1}{x}\ dx=\ln |x|+C\)
\(\int x^{-1}\ dx=\ln |x|+C\)
\(\int x^{-n}\ dx=\frac{1}{-n+1}x^{-n+1}+C,\quad n\ne 1\) \(\int x^{-5}\ dx=-\frac{1}{4}x^{-4}+C\)
\(\int \frac{1}{ax+b}\ dx=\frac{1}{a}\ln |ax+b|+C\) \(\int \frac{1}{2x+5}\ dx=\frac{1}{2}\ln |2x+5|+C\)
\(\int \cos x\ dx=\sin x + C\)
\(\int \sin x\ dx=-\cos x + C\)
\(\int \cos ax\ dx=\frac{1}{a}\sin ax + C\) \(\int \cos 3x\ dx=\frac{1}{3}\sin 3x + C\)
\(\int \sin ax\ dx=-\frac{1}{a}\cos ax + C\) \(\int \sin 5x\ dx=-\frac{1}{5}\cos 5x + C\)
\(\int \cot x=\ln |\sin x|+C\)
\(\int \tan x=-\ln |\cos x|+C\)
\(\int \frac{1}{\cos^{2} x}\ dx=\tan x + C\)
\(\int \frac{1}{\sin^{2} x}\ dx=-\cot x + C\)
\(\int \frac{1}{x^2+a^2}\ dx=\frac{1}{a}\arctan \frac{x}{a} + C\) \(\int \frac{1}{x^2+4}\ dx=\frac{1}{2}\arctan \frac{x}{2} + C\)
\(\int \frac{1}{x^2-a^2}\ dx=\frac{1}{2a} \ln \left |\frac{x-a}{x+a} \right| + C\) \(\int \frac{1}{x^2-9}\ dx=\frac{1}{6} \ln \left |\frac{x-3}{x+3} \right| + C\)
\(\int \frac{1}{\sqrt{a^2-x^2}}\ dx=\arcsin \frac{x}{a}+C\) \(\int \frac{1}{\sqrt{16-x^2}}\ dx=\arcsin \frac{x}{4}+C\)
\(\int \frac{1}{\sqrt{x^2+q}}\ dx=\ln \left |x+\sqrt{x^2+q} \right|+C\) \(\int \frac{1}{\sqrt{x^2+5}}\ dx=\ln \left |x+\sqrt{x^2+5} \right|+C\)
\(\int e^x\ dx= e^x+C\)
\(\int a^x\ dx= \frac{a^x}{\ln a}+C\) \(\int 2^x\ dx= \frac{2^x}{\ln 2}+C\)
\(\int \sinh x \ dx= \cosh x+C\)
\(\int \cosh x \ dx= \sinh x+C\)
\(\int \frac{1}{\sinh^{2} x} \ dx= -\coth x+C\)
\(\int \frac{1}{\cosh^{2} x} \ dx= \tanh x+C\)