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Supstitucija i parcijalna integracija     ODREĐENI INTEGRAL     Površina ravninskog lika


Nepravi integral

Izračunajte slijedeće integrale:

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

b)
$ \displaystyle\int\limits_{-\infty }^{\infty }\frac{ dx}{x^{2}+4x+5}
$ ,

c)
$ \displaystyle\int\limits_{-1}^{1}\frac{ dx}{x^{3}}$ .

Rješenje.

a)
Vrijedi

$\displaystyle \int\limits_{1}^{\infty }\frac{ dx}{x\sqrt{x^{2}+1}}$ $\displaystyle =\lim_{b\rightarrow \infty }\int\limits_{1}^{b}\frac{ dx}{x\sqrt{x^{2}+1}}$    
  $\displaystyle =\left\{ \begin{array}{cc} \sqrt{x^{2}+1}=t-x &  dx=\frac{4t^{2}...
...ne $t$ & $\sqrt{2}+1$ & $\sqrt{b^{2}+1}+b$ \end{tabular} \end{array} \right\}$    
  $\displaystyle =\lim_{b\rightarrow \infty }\int\limits_{\sqrt{2}+1}^{\sqrt{b^{2}...
...b}\frac{\frac{t^{2}+1}{2t^{2}} dt}{\frac{t^{2}-1}{2t}\cdot \frac{t^{2}+1}{2t}}$    
  $\displaystyle =\frac{1}{2}\lim_{b\rightarrow \infty }\int\limits_{\sqrt{2}+1}^{...
...tarrow \infty }\int\limits_{ \sqrt{2}+1}^{\sqrt{b^{2}+1}+b}\frac{ dt}{t^{2}-1}$    
  $\displaystyle =2\cdot \frac{1}{2}\lim_{b\rightarrow \infty }\ln \left\vert \frac{t-1}{t+1 }\right\vert \bigg\vert_{\sqrt{2}+1}^{\sqrt{b^{2}+1}+b}$    
  $\displaystyle =\lim_{b\rightarrow \infty }\ln \left\vert \frac{\sqrt{b^{2}+1}+b...
...}+b+1}\right\vert -\ln \left\vert \frac{\sqrt{2}+1-1}{\sqrt{2}+1+1} \right\vert$    
  $\displaystyle =\lim_{b\rightarrow \infty }\ln \left\vert \frac{\sqrt{\frac{1}{b...
...+\frac{1}{b}}\right\vert -\ln \left\vert \frac{\sqrt{2}}{\sqrt{2}+2}\right\vert$    
  $\displaystyle =\ln 1-\ln \frac{\sqrt{2}}{\sqrt{2}+2}=\ln \frac{\sqrt{2}+2}{\sqrt{2}}=\ln \left( 1+\sqrt{2}\right) .$    

b)
Vrijedi

$\displaystyle \int\limits_{-\infty }^{\infty }\frac{ dx}{x^{2}+4x+5}$ $\displaystyle =\lim_{a\rightarrow -\infty }\int\limits_{a}^{0}\frac{ dx}{x^{2}+4x+5}+\lim_{b\rightarrow \infty }\int\limits_{0}^{b}\frac{ dx}{x^{2}+4x+5}$    
  $\displaystyle =\lim_{a\rightarrow -\infty }\int\limits_{a}^{0}\frac{ dx}{\left...
...{b\rightarrow \infty }\int\limits_{0}^{b}\frac{ dx}{ \left( x+2\right) ^{2}+1}$    
  $\displaystyle =\lim_{a\rightarrow -\infty }\mathop{\mathrm{arctg}}\nolimits \le...
...w \infty }\mathop{\mathrm{arctg}}\nolimits \left( x+2\right) \bigg\vert_{0}^{b}$    
  $\displaystyle =\lim_{a\rightarrow -\infty }\left[ \mathop{\mathrm{arctg}}\nolim...
...m{arctg}}\nolimits \left( b+2\right) -\mathop{\mathrm{arctg}}\nolimits 2\right]$    
  $\displaystyle =\mathop{\mathrm{arctg}}\nolimits 2+\frac{\pi }{2}+\frac{\pi }{2}- \mathop{\mathrm{arctg}}\nolimits 2=\pi .$    

c)
Vrijedi

$\displaystyle \int\limits_{-1}^{1}\frac{ dx}{x^{3}}$ $\displaystyle =\int\limits_{-1}^{0}\frac{ dx}{x^{3}} +\int\limits_{0}^{1}\frac...
...+\lim_{\delta \rightarrow \infty }\int\limits_{0+\delta }^{1}\frac{ dx}{x^{3}}$    
  $\displaystyle =\lim_{\varepsilon \rightarrow 0}\frac{-1}{2x^{2}}\bigg\vert _{-1...
...}+\lim_{\delta \rightarrow \infty }\frac{-1}{2x^{2}} \bigg\vert_{0+\delta }^{1}$    
  $\displaystyle =\lim_{\varepsilon \rightarrow 0}\left( \frac{-1}{2\varepsilon ^{...
...m_{\delta \rightarrow 0}\left( \frac{-1}{2}+\frac{1}{ 2\varepsilon ^{2}}\right)$    
  $\displaystyle =\frac{1}{2}\lim_{\delta \rightarrow 0}\frac{1}{\delta ^{2}}-\frac{1}{2} \lim_{\varepsilon \rightarrow 0}\frac{1}{\varepsilon ^{2}}=\infty -\infty,$    

pa integral divergira.


Supstitucija i parcijalna integracija     ODREĐENI INTEGRAL     Površina ravninskog lika