# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 27/en

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Warm-up-exercises

### Exercise

Let

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

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

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

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

### Exercise

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

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

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

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

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

### Exercise

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

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

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

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

exists.

### Exercise

Formulate and prove computation rules for improper integrals (analogous to Lemma 23.5).

### Exercise

Decide whether the improper integral

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

exists.

### Exercise

Determine the improper integral

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

### Exercise

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

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

converges to this improper integral.

Hand-in-exercises

### Exercise (2 points)

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 (3 points)

Decide whether the improper integral

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

exists and compute it in case of existence.

### Exercise (5 points)

Give an example of a not bounded, continuous function

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

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

### Exercise (2 points)

Decide whether the improper integral

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

exists and compute it in case of existence.

### Exercise (4 points)

Decide whether the improper integral

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

exists.

### Exercise (8 points)

Decide whether the improper integral

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

exists.

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