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Integriranje iracionalnih funkcija racionalnom     NEODREĐENI INTEGRAL     Metoda neodređenih koeficijenata


Eulerova i trigonometrijska supstitucija

Izračunajte integrale:

a)
$ \displaystyle\int \frac{ dx}{1+\sqrt{x^{2}+2x+2}}$ ,

b)
$ \displaystyle\int \sqrt{3-2x-x^{2}} dx$ ,

Rješenje.

a)
Koristimo Eulerovu supstituciju [*][M2, poglavlje 1.7.2]. Vrijedi

$\displaystyle \int \frac{ dx}{1+\sqrt{x^{2}+2x+2}}$ $\displaystyle =\left\{ \begin{array}{l} \sqrt{x^{2}+2x+2}=t-x  x=\frac{t^{2}-...
...+2}   dx=\frac{t^{2}+2t+2}{2\left( t+1\right) ^{2}} dt \end{array} \right\}$    
  $\displaystyle =\int \frac{\frac{t^{2}+2t+2}{2\left( t+1\right) ^{2}}}{1+t-\frac...
...eft( t+1\right) ^{2}}}{\frac{ 2t+2+2t^{2}+2t-t^{2}+2}{2\left( t+1\right) }} dt$    
  $\displaystyle =\int \frac{\frac{t^{2}+2t+2}{t+1}}{t^{2}+4t+4} dt=\frac{t^{2}+2t+2}{( t+2) ^{2}( t+1) } dt$    
  $\displaystyle =\left\{ \begin{array}{c} \frac{t^{2}+2t+2}{\left( t+2\right) ^{2...
...}+\frac{C}{\left( t+2\right) ^{2}}  A=1,  B=0,  C=-2 \end{array} \right\}$    
  $\displaystyle =\int \frac{dt}{t+1}-2\int \frac{dt}{\left( t+2\right) ^{2}}$    
  $\displaystyle =\ln \left\vert t+1\right\vert -2\int \left( t+2\right) ^{-2}d\left( t+2\right)$    
  $\displaystyle =\ln \left\vert t+1\right\vert +2\left( t+2\right) ^{-1}+C$    
  $\displaystyle =\ln \left\vert \sqrt{x^{2}+2x+2}+x+1\right\vert$    
  $\displaystyle \quad +2\left( \sqrt{x^{2}+2x+2} +x+2\right) ^{-1}+C.$    

b)
Izraz pod korijenom nadpounjavamo do punog kvadrata, a zatim uvodimo dvije supstitucije:

$\displaystyle \int \sqrt{3-2x-x^{2}} dx$ $\displaystyle =\int \sqrt{4-\left( 1+x\right) ^{2}} dx=\left\{ \begin{array}{c} x+1=t   dx= dt \end{array} \right\}$    
  $\displaystyle =\int \sqrt{4-t^{2}} dt=\left\{ \begin{array}{c} t=2\sin z   dt=2\cos zdz \end{array} \right\}$    
  $\displaystyle =\int 2\cos z2\cos zdz=4\int \cos ^{2}zdz$    
  $\displaystyle =2\int \left( 1+2\cos z\right) dz=2\left( z+\frac{1}{2}\sin 2z\right) +C$    
  $\displaystyle =2\left( z+\sin z\sqrt{1-\sin ^{2}z}\right) +C$    
  $\displaystyle =2\arcsin \frac{t}{2}+t\sqrt{1-\frac{t^{2}}{4}}+C$    
  $\displaystyle =2\arcsin \frac{x+1}{2}+\left( x+1\right) \sqrt{1-\frac{\left( x+1\right) ^{2}}{4}}+C$    
  $\displaystyle =2\arcsin \frac{x+1}{2}+\frac{x+1}{2}\sqrt{3-2x-x^{2}}+C.$    


Integriranje iracionalnih funkcija racionalnom     NEODREĐENI INTEGRAL     Metoda neodređenih koeficijenata