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

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

Hint: Write the numerator polynomial using the denominator polynomial.

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?

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