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