Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 25

Warm-up-exercises

Exercise

Compute the definite integral

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

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

Exercise

Determine an antiderivative (primitive function) of the function

$\tan x.$

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

Determine an antiderivative (primitive function) of the function

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

Exercise

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

$a\longmapsto \int _{-1}^{2}at^{2}-a^{2}tdt$

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

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

Here, ${}x$ is the time in hours and ${}y=f(x)$ is the attention,n 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 ${}I$ be a real interval and let

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

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

$(bt+c)\cdot f(t).$

Exercise

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

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

using the antiderivative of ${}x^{n}$ and the theorem about integration of inverse function.

Exercise

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

Exercise

Let

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

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

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

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

Exercise

Compute by an appropriate substitution an antiderivative of

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

Exercise

Compute the definite integral of the function

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

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

Exercise

Compute the definite integral of the function

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

on ${}[1,4]$.

Hand-in-exercises

Exercise (3 marks)

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

${}f(t)={\begin{cases}t+1,{\text{ if }}0\leq t\leq 2\,,\\t^{2}-6t+11,{\text{ if }}2<t\leq 5\,,\\6,{\text{ if }}5<t\leq 6\,,\\-2t+18\,,{\text{ if }}6<t\leq 8\,.\end{cases}}\,$

Exercise (3 marks)

Determine an antiderivative (primitive function) of the function

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

Exercise (2 marks)

Determine an antiderivative (primitive function) of the function

$\arcsin x.$

Exercise (3 marks)

Determine an antiderivative (primitive function) of the function

$\sin(\ln x).$

Exercise (4 marks)

Determine an antiderivative (primitive function) of the function

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

Exercise (4 marks)

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

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

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

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