Osnovna pravila

  • $$\int f(x)dx=F(x)+C$$
  • $$F’(x)=f(x)$$
  • $$\int dx=x+C$$
  • $$\int k f(x)dx=k\int f(x)dx$$
  • $$\int (u+v+w)dx=\int u dx + \int v dx +\int w dx$$

Tablični integrali

  • $$\int x^ndx=\frac{x^{n+1}}{n+1}+C, n\neq -1$$
  • $$\int e^xdx=e^x+C$$
  • $$\int a^xdx=\int e^{x\ln a}dx=\frac{e^{x\ln a}}{\ln a}=\frac{a^x}{\ln a}+C, (a>0, a \neq 1)$$
  • $$\int \frac{dx}{x}=\ln \left | x \right |+C$$
  • $$\int \frac{dx}{1+x^2}=\arctan x+C$$
  • $$\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln \left |\frac{x-a}{x+a} \right |+C$$
  • $$\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\ln \left |\frac{a+x}{a-x} \right |+C$$
  • $$\int \frac{dx}{\sqrt{x^2\pm a^2}}=\ln (x+\sqrt{x^2\pm a^2})+C$$
  • $$\int \frac{dx}{\sqrt{a^2-x^2}}=\arcsin \frac{x}{a}+C,\; \; a>0$$
  • $$\int \sin xdx=-\cos x+C$$
  • $$\int \cos xdx=\sin x+C$$
  • $$\int \tan xdx=-\ln \cos x+C$$
  • $$\int \cot xdx=\ln \sin x+C$$
  • $$\int \frac{dx}{\sin ^2x}=-\cot x+C$$
  • $$\int \frac{dx}{\cos ^2x}=\tan x+C$$

Metoda smene

Uputstva

  • $$\int F(ax+b)dx=\frac{1}{2}\int F(t)dt, t=ax+b$$
  • $$\int F(\sqrt{ax+b})dx=\frac{2}{a}\int tF(t)dt, t=\sqrt{ax+b}$$
  • $$\int F(\sqrt[n]{ax+b})dx=\frac{n}{a}\int t^{n-1}F(t)dt,\; \; t=\sqrt[n]{ax+b}$$
  • $$\int F(\sqrt{a^2-x^2})dx=a\int F(a\cos t)\cos tdt,\; \; x=a\sin t$$
  • $$\int F(e^{ax})dx=\frac{1}{a}\int \frac{F(t)}{t}dt, \; \; t=e^{ax}$$
  • $$\int F(\ln x)dx=\int F(t)e^tdt,\; \; t=\ln x$$

Nekoliko primera

  • $$\int (ax+b)^ndx=\frac{(ax+b)^{n+1}}{a(n+1)}+C$$
  • $$\int \frac{dx}{ax+b}=\frac{1}{a}\ln \left | ax+b \right |+C$$
  • $$\int \frac{ax+b}{cx+d}dx=\frac{a}{c}x+\frac{bc-ad}{c^2}\ln \left | cx+d \right |+C$$
  • $$\int \frac{dx}{a^2+x^2}=\frac{1}{a}\arctan \frac{x}{a}+C$$
  • $$\int \frac{dx}{(x+a)(x+b)}=\frac{1}{a-b}\ln \left |\frac{x+b}{x+a} \right |+C, a\neq b$$
  • $$\int \frac{xdx}{(x+a)(x+b)}=$$ $$=\frac{1}{a-b}(a\ln |x+a|-b\ln |x+b|)+C, \; \; a\neq b$$
  • $$\int \frac{xdx}{x^2-a^2}=\frac{1}{2}\ln |x^2-a^2|+C$$
  • $$\int \frac{xdx}{x^2+a^2}=\frac{1}{2}\ln |x^2+a^2|+C$$
  • $$\int \frac{dx}{(x^2+a^2)^2}=$$ $$=\frac{1}{2a^2}\cdot \frac{x}{x^2+a^2}+\frac{1}{2a^3}\arctan \frac{x}{a}+C$$
  • $$\int \frac{xdx}{(x^2+a^2)^2}=-\frac{1}{2}\cdot \frac{1}{x^2+a^2}+C$$
  • $$\int \frac{dx}{(x^2+a^2)(x+b)}=$$ $$=\frac{1}{a^2+b^2}\left ( \ln \frac{|x+b|}{\sqrt{x^2+a^2}}+\frac{b}{a}\arctan \frac{x}{a} \right )+C$$
  • $$\int \frac{xdx}{(x^2+a^2)(x+b)}=$$ $$=\frac{1}{a^2+b^2}\left ( \arctan \frac{x}{a} -b\ln \frac{|x+b|}{\sqrt{x^2+a^2}}\right )+C$$

Integrali iracionalnih funkcija

\(\int \frac{dx}{\sqrt{ax+b}}=\frac{2}{a}\sqrt{ax+b}+C\)

\(\int \sqrt{ax+b}dx=\frac{2}{3a}(ax+b)^{\frac{3}{2}}+C\)

\(\int \frac{xdx}{\sqrt{ax+b}}=\frac{2(ax-2b)}{3a^2}\sqrt{ax+b}+C\)

\(\int x\sqrt{ax+b}dx=\frac{2(3ax-2b)}{15a^2}(ax+b)^{\frac{2}{3}}+C\)

\(\int x^2\sqrt{a+bx}dx=\frac{2 \sqrt{(a+bx)^3}(8a^2-12abx+15b^2x^2)}{105b^3}\)

Integrali trigonometrijskih funkcija

\(\int \sin ^2xdx=\frac{x}{2}-\frac{1}{4}\sin 2x+C\)

\(\int \cos ^2xdx=\frac{x}{2}+\frac{1}{4}\sin 2x+C\)

\(\int \tan ^2xdx=\tan x-x+C\)

\(\int \cot ^2xdx=-\cot x-x+C\)

\(\int \sin ^3xdx=\frac{1}{2}\cos ^3x-\cos x+C\)

\(\int \cos ^3xdx=\sin x-\frac{1}{3}\sin ^3x+C\)

Parcijalna integracija

\(\int udv=uv-\int vdu\)

Određeni integrali

\(\int_{a}^{b} f(x)dx=F(x)\Big |_{a}^{b}=F(b)-F(a)\)

\(\int_{a}^{b}(f(x)\pm g(x))dx=\int_{a}^{b}f(x)dx\pm \int_{a}^{b} g(x)dx=\)

\(\int_{a}^{b}kf(x)dx=k\int_{a}^{b}f(x)dx\)

\(\int_{a}^{a}f(x)dx=0\)

\(\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx\)

\(\int_{a}^{b}f(x)dx=\int_{a}^{c}f(x)dx+\int_{c}^{b}f(x)dx, \; \; a<c<b\)

Primena određenog integrala

Površina ravne figure: $$P(F)=\int_{a}^{b}f(x)dx$$

Zapremina obrtnog tela: $$V(\Phi )=\pi \int_{a}^{b}f^2(x)dx$$

Dužina luka krive: $$l=\int_{a}^{b}\sqrt{1+f'^2(x)}dx$$