Wzór funkcji Całka funkcji $$f(x) = a$$ $$\int a \: dx = ax + C$$ $$f(x) = x$$ $$\int x dx = \dfrac{1}{2} x^2 + C$$ $$f(x) = x^n$$ $$\int x^n dx = \dfrac{1}{n + 1} x^{n+1} + C$$, dla $$n \neq -1$$ $$f(x) =\dfrac{1}{x}$$ $$\int \dfrac{1}{x} dx =ln\left | x \right | + C$$ $$f(x) =a^x$$ $$\int a^x \: dx = \dfrac{1}{ln \: a}a^x + C$$ $$f(x) =ln \: x$$ $$\int ln \: x \: dx = (x-1) \: ln \: x + C$$ $$f(x) =log_a x$$ $$\int \: log_a x\ dx = \dfrac{x}{ln \: a}(ln \: x - 1) + C$$ $$f(x) =e^x$$ $$\int e^x \: dx = e^x + C$$ $$f(x) =\sqrt{x}$$ $$\int \sqrt{x} \: dx = \dfrac{2}{3} \sqrt{x^3} + C$$ $$f(x) =\dfrac{1}{\sqrt{x}}$$ $$\int \dfrac{1}{\sqrt{x}} \: dx =2 \sqrt{x} + C$$ $$f(x) =\dfrac{1}{ax +b}$$ $$\int \dfrac{1}{ax +b} dx = \dfrac{1}{a} ln \left |ax +b \right |+ C$$, dla $$a \neq 0$$ $$f(x) = sin \: x$$ $$\int sin \: x \: dx = - cos \: x + C$$ $$f(x) = cos \: x$$ $$\int cos \: x \: dx = sin \: x + C$$ $$f(x) = tg \: x$$ $$\int \: tg \: x \: dx = -ln \left| cos \: x \right|+ C$$ $$f(x) = ctg \: x$$ $$\int \: ctg \: x \: dx = ln \left| sin \: x \right|+ C$$ $$f(x) = \dfrac{1}{cos^2 x}$$ $$\int \: \dfrac{1}{cos^2 x} \: x \: dx = tg \: x + C$$, gdy $$cos \: x \neq 0$$ $$f(x) = \dfrac{1}{sin^2 x}$$ $$\int \: \dfrac{1}{sin^2 x} \: x \: dx =-ctg \: x + C$$, gdy $$sin \: x \neq 0$$ $$f(x) = \dfrac{1}{x^2 +a^2}$$ $$\int \: \dfrac{1}{x^2 + a^2} dx =\dfrac{1}{a} arc \: tg \dfrac{x}{a} + C$$, dla $$a \neq 0$$ $$f(x) = \dfrac{1}{\sqrt{a^2 - x^2}}$$ $$\int \: \dfrac{1}{\sqrt{a^2 - x^2}}dx = arc \: sin \dfrac{x}{a} + C$$, dla $$a \neq 0$$ $$f(x) = \dfrac{1}{\sqrt{x^2 - a^2}}$$ $$\int \:\dfrac{1}{\sqrt{x^2 - a^2}} dx = ln \left| x+ \sqrt{x^2 - a^2} \right | + C$$ $$f(x) = (ax + b)^n$$ $$\int \: (ax + b)^n dx = \dfrac{1}{a(n+1)} (ax + b)^{n+1} + C$$ dla $$n \neq -1$$ $$f(x) = \dfrac{1}{a^2 - x^2}$$ $$\int \dfrac{1}{a^2 - x^2} dx = \dfrac{1}{2a}ln \left|\dfrac{a+x}{a-x} \right| +C$$, dla $$a>0 \: i \: \left|x \right| \neq a$$
$$\int ln \: x \: dx = (x-1) \: ln \: x + C$$