Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 27

Warm-up-exercises

Exercise

Let

$f\colon \mathbb {R} \longrightarrow \mathbb {R}$

be an increasing function and ${}b\in \mathbb {R}$. Show that the sequence ${}f(n)$, ${}n\in \mathbb {N}$, converges to ${}b$ if and only if

${}\operatorname {lim} _{x\rightarrow +\infty }\,f(x)=b\,$

holds, i.e. if the limit of the function for ${}x\rightarrow +\infty$ is ${}b$.

Exercise

Let ${}I$ be an interval, ${}r$ a boundary point of ${}I$ and

$f\colon I\longrightarrow \mathbb {R}$

a continuous function. Prove that the existence of the improper integral

$\int _{a}^{r}f(t)dt$

does not depend on the choice of the starting point ${}a\in I$.

Exercise

Let

${}I={]r,s[}$ be a bounded open interval and

$f\colon I\longrightarrow \mathbb {R}$

a continuous function, which can be extended continuously to ${}[r,s]$. Prove that the improper integral

$\int _{r}^{s}f(t)dt$

exists.

Exercise

Formulate and prove computation rules for improper integrals (analogous to

the rules of definite integrals.

Exercise

Decide whether the improper integral

$\int _{1}^{\infty }{\frac {x^{2}-3x+5}{x^{4}+2x^{3}+5x+8}}dx$

exists.

Exercise

Determine the improper integral

$\int _{0}^{\infty }e^{-t}dt.$

Exercise

Let ${}I=[a,b]$ be a bounded interval and let

$f\colon ]a,b[\longrightarrow \mathbb {R}$

be a continuous function. Let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a decreasing sequence in ${}I$ with limit ${}a$ and let ${}{\left(y_{n}\right)}_{n\in \mathbb {N} }$ be an increasing sequence in ${}I$ with limit ${}b$. Assume that the improper integral ${}\int _{a}^{b}f(t)dt$ exists. Prove that the sequence

${}w_{n}=\int _{x_{n}}^{y_{n}}f(t)dt\,$

converges to this improper integral.

Hand-in-exercises

Exercise

Compute the energy that would be necessary to move the Earth, starting from the current position relative to the Sun, infinitely far away from the Sun.

Exercise

Decide whether the improper integral

$\int _{0}^{\infty }{\frac {1}{(x+1){\sqrt {x}}}}dx$

exists and compute it in case of existence.

Exercise

Give an example of a not bounded, continuous function

$f\colon \mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},$

such that the improper integral ${}\int _{0}^{\infty }f(t)dt$ exists.

Exercise

Decide whether the improper integral

$\int _{-1}^{1}{\frac {1}{\sqrt {1-t^{2}}}}dt$

exists and compute it in case of existence.

Exercise

Decide whether the improper integral

$\int _{1}^{\infty }{\frac {x^{3}-3x+5}{x^{4}+2x^{3}+5x+8}}dx$

exists.

Exercise

Decide whether the improper integral

$\int _{0}^{\infty }{\frac {\sin x}{x}}dx$

exists.

(Do not try to find an antiderivative for the integrand.)