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

$f:I⟶ℝ$

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

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

Exercise

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

${ℝ}_{+}⟶{ℝ}_{+},x⟼x^{1/n},$

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

$f:[a,b]⟶[c,d]$

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

$f:{ℝ}_{+}⟶ℝ,x⟼f(x)=\sqrt{x}-\frac{1}{\sqrt{x}}+\frac{1}{2x+3}-e^{-x},$

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 $a,b\in {ℝ}_{+}$, 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

$f:I⟶ℝ$

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

$(a{t}^2+bt+c)\cdot f(t).$

Exercise (5 marks)

Let

$f:[0,1]⟶{ℝ}_{+}$

be a differentiable function with $f^{\prime }(x)>0$ 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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