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\begin{document}
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Institutionen för Matematik   \newline
KTH  \newline
Lars Filipsson \newline

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\centerline{ \bf  Uppgifter till räkneövningen den 28 november }
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\centerline{ 5B1132 Amelia 1 för P ht 2003}
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Uppgifterna 1-4 är tänkta att räknas av lärare på tavlan under max 45 minuter. Resten av uppgifterna ska teknologerna lösa själva, individuellt eller i grupp, under handledning av läraren.

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\problem{Bestäm arean av området mellan $x$-axeln och kurvan $y=x\cos{x}$, \newline $0\le x \le \pi /2$.}

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\problem{Beräkna integralen $\displaystyle \int_0^1 \frac{\arctan{x}}{1+x^2}\, dx.$
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\problem{Bestäm en primitiv funktion till  $\arctan{x}$.}

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\problem{Är funktionen $f(x)=e^{x^2}$ integrerbar på intervallet $[0,1]$? Om ja, beräkna exakt eller approximativt 

$$\int_0^1 e^{x^2}\, dx.$$
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\problem{Bestäm arean mellan $x$-axeln och kurvan $\displaystyle y=\frac{1}{x\sqrt{x}}$, $1\le x \le 4$.}

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\problem{Beräkna integralen $\displaystyle \int_1^4 \frac{e^{\sqrt{x}}}{\sqrt{x}} \, dx.$
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\problem{Bestäm en primitiv funktion till $xe^{x}$.}

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\problem{Beräkna integralen $\displaystyle \int_0^{\pi^2 /4} \sin{\sqrt{x}}\, dx.$}

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\problem{Bestäm arean mellan kurvan $\displaystyle y=\tan^2 x$, $0\le x \le \pi /4$,  och $x$-axeln, .
}

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\problem{Beräkna integralen $\displaystyle \int \frac{2x+2}{x^2+2x+2}\, dx$.}

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\problem{Beräkna integralen $\displaystyle \int \frac{1}{x^2+2x+2}\, dx$.}

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\problem{Beräkna integralen $\displaystyle \int \frac{2x+1}{x^2+2x+2}\, dx$.}







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