# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 20

    Exercises

### Exercise

Show by induction over $n$, using integration by parts, that
    $\int _{0}^{1}x^{m}(1-x)^{n}dx={\frac {m!n!}{(m+n+1)!}}\,$
holds.
The following exercises are about determining primitive functions. This includes the choice of suitable domains for the functions.

### Exercise

Determine an antiderivative (primitive function) of the function
    $x^{n}\cdot \ln x.$

### Exercise

Determine an antiderivative (primitive function) of the function
    $\tan x.$

### 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 an antiderivative (primitive function) of the function
    ${\frac {1+3{\sqrt[{6}]{x-2}}}{{\sqrt[{3}]{(x-2)^{2}}}-{\sqrt {x-2}}}}.$

### Exercise

Determine an antiderivative (primitive function) of the function
    $(\ln(1+\sin x))\cdot \sin x.$

### 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 the definite integral
    $\int _{0}^{\sqrt {\pi }}x\sin x^{2}dx$

### 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 $[1,4]$.

### Exercise

Compute by an appropriate substitution an antiderivative of
    ${\sqrt {3x^{2}+5x-4}}.$

### Exercise

Prove the relationship
    $\int _{1}^{ab}{\frac {1}{x}}dx=\int _{1}^{a}{\frac {1}{x}}dx+\int _{1}^{b}{\frac {1}{x}}dx\,$
for $a,b\in \mathbb {R} _{+}$, only using rules for integration.

### Exercise

Determine the areas of the regions, surrounded by the blue curves, sketched on the right.
    Hand-in-exercises

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

### Exercise (5 marks)

Let
    $f\colon [0,1]\longrightarrow \mathbb {R} _{+}$
be a differentiable function with $f'(x)>0$ for all $x>0$. For what points $t\in [0,1]$ does the area of the hatched surface have a local extremum? Is it a minimum or a maximum?