Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 25/en

Zur Navigation springen Zur Suche springen

Warm-up-exercises

Exercise

Compute the definite integral

${\displaystyle \int _{0}^{\sqrt {\pi }}x\sin x^{2}\,dx.}$

In the following exercises, which involve the determination of antiderivative functions, consider an appropriate domain of definition.

Exercise

Determine an antiderivative of the function

${\displaystyle \tan x.}$

Exercise

Determine an antiderivative of the function

${\displaystyle x^{n}\cdot \ln x.}$

Exercise

Determine an antiderivative of the function

${\displaystyle e^{\sqrt {x}}.}$

Exercise

Determine an antiderivative of the function

${\displaystyle {\frac {x^{3}}{\sqrt[{5}]{x^{4}+2}}}.}$

Exercise

Determine an antiderivative of the function

${\displaystyle {\frac {\sin ^{2}x}{\cos ^{2}x}}.}$

Exercise

Determine for which ${\displaystyle {}a\in \mathbb {R} }$ the function

${\displaystyle a\longmapsto \int _{-1}^{2}at^{2}-a^{2}t\,dt}$

has a maximum or a minimum.

Exercise

According to recent studies the student's attention skills during the day are described by the following function

${\displaystyle [8,18]\longrightarrow \mathbb {R} ,\,x\longmapsto f(x)=-x^{2}+25x-100.}$

Here ${\displaystyle {}x}$ is the time in hours and ${\displaystyle {}y=f(x)}$ is the attention measured in micro-credit points per second. When should one start a one and a half hour lecture, such that the total attention skills are optimal? How many micro-credit points will be added during this lecture?

Exercise

Let ${\displaystyle {}I}$ be a real interval and let

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

be a continuous function with antiderivative ${\displaystyle {}F}$. Let ${\displaystyle {}G}$ be an antiderivative of ${\displaystyle {}F}$ and let ${\displaystyle {}b,c\in \mathbb {R} }$. Determine an antiderivative of the function

${\displaystyle (bt+c)\cdot f(t).}$

Exercise

Let ${\displaystyle {}n\in \mathbb {N} _{+}}$. Determine an antiderivative of the function

${\displaystyle \mathbb {R} _{+}\longrightarrow \mathbb {R} _{+},\,x\longmapsto x^{1/n},}$

using the antiderivative of ${\displaystyle {}x^{n}}$ and Theorem 25.4.

Exercise

Determine an antiderivative of the natural logarithm function using the antiderivative of its inverse function.

Exercise

Let

${\displaystyle f\colon [a,b]\longrightarrow [c,d]}$

be a bijective, continuous differentiable function. Prove the formula for the antiderivative of the inverse function by the integral

${\displaystyle \int _{a}^{b}f^{-1}(y)dy}$

using the substitution ${\displaystyle {}y=f(x)}$ and then integration by parts.

Exercise

Compute by an appropriate substitution an antiderivative of

${\displaystyle {\sqrt {3x^{2}+5x-4}}.}$

Exercise

Compute the definite integral of the function

${\displaystyle f\colon \mathbb {R} \longrightarrow \mathbb {R} ,\,x\longmapsto f(x)=2x^{3}+3e^{x}-\sin x,}$

on ${\displaystyle {}[-1,0]}$.

Exercise

Compute the definite integral of the function

${\displaystyle f\colon \mathbb {R} _{+}\longrightarrow \mathbb {R} ,\,x\longmapsto f(x)={\sqrt {x}}-{\frac {1}{\sqrt {x}}}+{\frac {1}{2x+3}}-e^{-x},}$

on ${\displaystyle {}[1,4]}$.

Hand-in-exercises

Exercise (4 points)

Compute the definite integral ${\displaystyle {}\int _{0}^{8}f(t)\,dt}$, where the function ${\displaystyle {}f}$ is

${\displaystyle f(t)={\begin{cases}t+1,{\text{ if }}0\leq t\leq 2\,,\\t^{2}-6t+11,{\text{ if }}2

Exercise (3 points)

Determine an antiderivative of the function

${\displaystyle x^{3}\cdot \cos x-x^{2}\cdot \sin x.}$

Exercise (2 points)

Determine an antiderivative of the function

${\displaystyle \arcsin x.}$

Exercise (4 points)

Determine an antiderivative of the function

${\displaystyle \sin(\ln x).}$

Exercise (5 points)

Determine an antiderivative of the function

${\displaystyle e^{x}\cdot {\frac {x^{2}+1}{(x+1)^{2}}}.}$

Exercise (5 points)

Let ${\displaystyle {}I}$ be a real interval and let

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

be a continuous function with antiderivative ${\displaystyle {}F}$. Let ${\displaystyle {}G}$ be an antiderivative of ${\displaystyle {}F}$ and ${\displaystyle {}H}$ an antiderivative of ${\displaystyle {}G}$. Let ${\displaystyle {}a,b,c\in \mathbb {R} }$. Determine an antiderivative of the function

${\displaystyle (at^{2}+bt+c)\cdot f(t).}$