Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 26

Warm-up-exercises

Exercise

Determine the partial fraction decomposition of

${\frac {3X^{5}+4X^{4}-2X^{2}+5X-6}{X^{3}}}.$

Exercise

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

Exercise

Determine the complex and the real partial fraction decomposition of

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

Exercise

Determine the complex partial fraction decomposition of

${\frac {1}{X^{3}-1}}.$

Exercise

Determine the complex and the real partial fraction decomposition of

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

Exercise

Determine the complex and the real partial fraction decomposition of

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

Exercise

Determine the complex and the real partial fraction decomposition of

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative of the function

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

through partial fraction decomposition.

Exercise

We consider the function

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

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

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

Exercise

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

Hand-in-exercises

Exercise

Write the rational function

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

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

Exercise

Determine the complex and the real partial fraction decomposition of

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

Exercise

Determine the complex and the real partial fraction decomposition of

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative (primitive function) of the function

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