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

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Warm-up-exercises

### Exercise

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

### Exercise

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

### Exercise

Determine the complex and the real partial fraction decomposition

of
${\displaystyle {\frac {1}{X^{2}(X^{2}+1)}}.}$

### Exercise

Determine the

complex partial fraction decomposition of
${\displaystyle {\frac {1}{X^{3}-1}}.}$

### Exercise

Determine the complex and the real partial fraction decomposition

of
${\displaystyle {\frac {1}{X^{3}(X-1)^{3}}}.}$

### Exercise

Determine the complex and the real partial fraction decomposition

of
${\displaystyle {\frac {X^{3}+4X^{2}+7}{X^{2}-X-2}}.}$

### Exercise

Determine the complex and the real partial fraction decomposition

of
${\displaystyle {\frac {X^{7}+X^{4}-5X+3}{X^{8}+X^{6}-X^{4}-X^{2}}}.}$

### Exercise

Determine an antiderivative of the function

${\displaystyle {\frac {1}{x^{2}+5}}.}$

### Exercise

Determine an antiderivative of the function

${\displaystyle {\frac {1}{x^{2}-5}}.}$

### Exercise

Determine an antiderivative of the function

${\displaystyle {\frac {1}{2x^{2}+x-1}}.}$

### Exercise

Determine an antiderivative of the function

${\displaystyle {\frac {5x^{3}+4x-3}{x^{2}+1}}}$

through partial fraction decomposition.

### Exercise

We consider the function

${\displaystyle f\colon \mathbb {R} \setminus \{1\}\longrightarrow \mathbb {R} ,\,x\longmapsto {\frac {x^{5}+3x^{3}-2x^{2}+x-1}{(x-1)^{2}(x^{2}+1)}}.}$

a) Determine the real partial fraction decomposition of ${\displaystyle {}f}$.

b) Determine an antiderivative of ${\displaystyle {}f}$ for ${\displaystyle {}x>1}$.

### Exercise

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

Hand-in-exercises

### Exercise (4 points)

Write the rational function

${\displaystyle {\frac {2x^{3}-4x^{2}+5x-1}{4x+3}}}$

in the new variables ${\displaystyle {}u=4x+3}$. Compute the antiderivative through the real partial fraction decomposition and through the substitution ${\displaystyle {}u=4x+3}$.

### Exercise (4 points)

Determine the complex and the real partial fraction decomposition

of
${\displaystyle {\frac {1}{X^{4}-1}}.}$

### Exercise (4 points)

Determine the complex and the real partial fraction decomposition

of
${\displaystyle {\frac {1}{X(X-1)(X-2)(X-3)}}.}$

### Exercise (4 points)

Determine an antiderivative of the function

${\displaystyle {\frac {1}{1+x^{4}}}.}$

### Exercise (5 points)

Determine an antiderivative of the function

${\displaystyle {\frac {3x-5}{(x^{2}+2x+7)^{2}}}.}$

### Exercise (1 point)

Determine an antiderivative of the function

${\displaystyle {\frac {7x^{6}-18x^{5}+8x^{3}-9x^{2}+2}{x^{7}-3x^{6}+2x^{4}-3x^{3}+2x-5}}.}$