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

Exercises

### Exercise

Show by induction over ${\displaystyle {}n}$, using integration by parts, that

${\displaystyle {}\int _{0}^{1}x^{m}(1-x)^{n}dx={\frac {m!n!}{(m+n+1)!}}\,}$

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

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

### Exercise

Determine an antiderivative (primitive function) of the function

${\displaystyle \tan x.}$

### Exercise

Determine an antiderivative (primitive function) of the function

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

### Exercise

Determine an antiderivative (primitive function) of the function

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

### Exercise

Determine an antiderivative (primitive function) of the function

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

### Exercise

Determine an antiderivative (primitive function) of the function

${\displaystyle {\frac {1+3{\sqrt[{6}]{x-2}}}{{\sqrt[{3}]{(x-2)^{2}}}-{\sqrt {x-2}}}}.}$

### Exercise

Determine an antiderivative (primitive function) of the function

${\displaystyle (\ln(1+\sin x))\cdot \sin x.}$

### 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 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

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

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 the definite integral

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

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

### Exercise

Compute by an appropriate substitution an antiderivative of

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

### Exercise

Prove the relationship

${\displaystyle {}\int _{1}^{ab}{\frac {1}{x}}dx=\int _{1}^{a}{\frac {1}{x}}dx+\int _{1}^{b}{\frac {1}{x}}dx\,}$

for ${\displaystyle {}a,b\in \mathbb {R} _{+}}$, 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

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

### Exercise (2 marks)

Determine an antiderivative (primitive function) of the function

${\displaystyle \arcsin x.}$

### Exercise (3 marks)

Determine an antiderivative (primitive function) of the function

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

### Exercise (4 marks)

Determine an antiderivative (primitive function) of the function

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

Hint: Write the numerator polynomial using the denominator polynomial.

### Exercise (4 marks)

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

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

### Exercise (5 marks)

Let

${\displaystyle f\colon [0,1]\longrightarrow \mathbb {R} _{+}}$

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