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

Warm-up-exercises

Determine the partial fraction decomposition of

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

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

Determine the complex and the real partial fraction decomposition of

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

Determine the complex partial fraction decomposition of

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

Determine the complex and the real partial fraction decomposition of

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

Determine the complex and the real partial fraction decomposition of

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

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}}}.

Determine an antiderivative (primitive function) of the function

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

Determine an antiderivative (primitive function) of the function

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

Determine an antiderivative (primitive function) of the function

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

Determine an antiderivative of the function

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

through partial fraction decomposition.

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 ${}x>1$ .

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

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 ${}u=4x+3$ .

Determine the complex and the real partial fraction decomposition of

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

Determine the complex and the real partial fraction decomposition of

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

Determine an antiderivative (primitive function) of the function

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

Determine an antiderivative (primitive function) of the function

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

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}}.