Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 26/en/latex

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\setcounter{section}{26}






\zwischenueberschrift{Warm-up-exercises}




\inputexercise
{}
{

Determine the partial fraction decomposition of
\mathdisp {{ \frac{ 3X^5+4X^4-2X^2+5X-6 }{ X^3 } }} { . }

}
{} {}




\inputexercise
{}
{

Determine the coefficients in the partial fraction decomposition of Example 26.5 by replacing $X$ with some numbers.

}
{} {}




\inputexercise
{}
{

Determine the \definitionsverweis {complex}{}{} and the \definitionsverweis {real partial fraction decomposition}{}{} of
\mathdisp {\frac{ 1 }{ X^2(X^2+1) }} { . }

}
{} {}




\inputexercise
{}
{

Determine the \definitionsverweis {complex partial fraction decomposition}{}{} of
\mathdisp {\frac{ 1 }{ X^3-1 }} { . }

}
{} {}




\inputexercise
{}
{

Determine the \definitionsverweis {complex}{}{} and the \definitionsverweis {real partial fraction decomposition}{}{} of
\mathdisp {\frac{ 1 }{ X^3(X-1)^3 }} { . }

}
{} {}




\inputexercise
{}
{

Determine the \definitionsverweis {complex}{}{} and the \definitionsverweis {real partial fraction decomposition}{}{} of
\mathdisp {\frac{ X^3+4X^2+7 }{ X^2-X-2 }} { . }

}
{} {}




\inputexercise
{}
{

Determine the \definitionsverweis {complex}{}{} and the \definitionsverweis {real partial fraction decomposition}{}{} of
\mathdisp {\frac{ X^7+X^4-5X+3 }{ X^8+X^6-X^4-X^2 }} { . }

}
{} {}




\inputexercise
{}
{

Determine an \definitionsverweis {antiderivative}{}{} of the \definitionsverweis {function}{}{}
\mathdisp {{ \frac{ 1 }{ x^2+5 } }} { . }

}
{} {}




\inputexercise
{}
{

Determine an \definitionsverweis {antiderivative}{}{} of the \definitionsverweis {function}{}{}
\mathdisp {{ \frac{ 1 }{ x^2-5 } }} { . }

}
{} {}




\inputexercise
{}
{

Determine an \definitionsverweis {antiderivative}{}{} of the \definitionsverweis {function}{}{}
\mathdisp {{ \frac{ 1 }{ 2x^2+x-1 } }} { . }

}
{} {}




\inputexercise
{}
{

Determine an antiderivative of the function
\mathdisp {{ \frac{ 5x^3+4x-3 }{ x^2+1 } }} { }
through partial fraction decomposition.

}
{} {}




\inputexercise
{}
{

We consider the function \maabbeledisp {f} {\R \setminus \{1\}} { \R } {x} {{ \frac{ x^5+3x^3-2x^2+x-1 }{ (x-1)^2(x^2+1) } } } {.}

a) Determine the real partial fraction decomposition of $f$.

b) Determine an antiderivative of $f$ for $x>1$.

}
{} {}




\inputexercise
{}
{

Find a representation of the \definitionsverweis {rational number}{}{}
\mathl{1/60}{} as a sum of rational numbers, such that every denominator is a power of a prime number.

}
{} {}






\zwischenueberschrift{Hand-in-exercises}




\inputexercise
{4}
{

Write the rational function
\mathdisp {{ \frac{ 2x^3-4x^2+5x-1 }{ 4x+3 } }} { }
in the new variables
\mathl{u=4x+3}{.} Compute the antiderivative through the real partial fraction decomposition and through the substitution
\mathl{u=4x+3}{.}

}
{} {}




\inputexercise
{4}
{

Determine the \definitionsverweis {complex}{}{} and the \definitionsverweis {real partial fraction decomposition}{}{} of
\mathdisp {\frac{ 1 }{ X^4-1 }} { . }

}
{} {}




\inputexercise
{4}
{

Determine the \definitionsverweis {complex}{}{} and the \definitionsverweis {real partial fraction decomposition}{}{} of
\mathdisp {\frac{ 1 }{ X(X-1)(X-2)(X-3) }} { . }

}
{} {}




\inputexercise
{4}
{

Determine an \definitionsverweis {antiderivative}{}{} of the \definitionsverweis {function}{}{}
\mathdisp {{ \frac{ 1 }{ 1+x^4 } }} { . }

}
{} {}




\inputexercise
{5}
{

Determine an \definitionsverweis {antiderivative}{}{} of the \definitionsverweis {function}{}{}
\mathdisp {{ \frac{ 3x-5 }{ (x^2+2x+7)^2 } }} { . }

}
{} {}




\inputexercise
{1}
{

Determine an \definitionsverweis {antiderivative}{}{} of the \definitionsverweis {function}{}{}
\mathdisp {{ \frac{ 7x^6-18x^5+8x^3-9x^2+2 }{ x^7-3x^6+2x^4-3x^3+2x-5 } }} { . }

}
{} {}



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