1103067901
Level:
A
Find the area of the yellow region indicated in the picture.
\( 6 \)
\( 0 \)
\( 3 \)
\( -6 \)
A
1103068103
Level:
A
Find the area of the yellow region indicated in the picture.
\( 2\sqrt2 \)
\( 2\sqrt3 \)
\( \sqrt2 \)
\( 4\sqrt2 \)
1103068102
Level:
A
Find the area of the yellow triangle ABC indicated in the picture. Read all the needed values from the picture.
\( 10 \)
\( 11 \)
\( 9.5 \)
\( 10.5 \)
1103068101
Level:
A
Find the area of the yellow region that is bounded by the parabola and the line as is indicated in the picture. Read all the needed values from the picture.
\( 4.5 \)
\( 22.5 \)
\( 3.75 \)
\( 10.25 \)
1003018710
Level:
A
Four children evaluated the following integral \( I \) on \( (0;\infty) \). Who made a mistake?
\[ I =\int\left(\frac18\sqrt[8]{x^3}+\frac12\sqrt{x^9}-\frac15\sqrt[5]{x^6} \right)\mathrm{d}x \]
Paul: \( I =\frac1{11}\left(x^3\sqrt[8]x+x\sqrt{x^5}-x\sqrt[5]{x^2}\right)+c\text{, }c\in\mathbb{R} \)
Jane: \( I =\frac1{11}\left(x\sqrt[8]{x^3}+x^5\sqrt x-x^2\sqrt[5] x+c\right)\text{, }c\in\mathbb{R} \)
Ann: \( I =\frac1{11}\left(x\sqrt[8]{x^3}+x^5\sqrt x-x^2\sqrt[5]x\right)+c\text{, }c\in\mathbb{R} \)
Miky: \( I =\frac x{11}\sqrt[8]{x^3}+\frac{x^5}{11}\sqrt x-\frac{x^2}{11}\sqrt[5]x+c\text{, }c\in\mathbb{R} \)
1003018709
Level:
A
Four children evaluated the following integral \( I \) on \( \mathbb{R} \). Who made a mistake?
\[ I =\int\left(5x^4+9x^2-6x\right)\mathrm{d}x \]
Paul: \( I =\left(x^3+3\right)\cdot x^2-3x^3+c\text{, }c\in\mathbb{R} \)
Jane: \( I =x^2\left(x^3+3x-3\right)+c\text{, }c\in\mathbb{R} \)
Ann: \( I =x^5+3x^3-3x^2+c\text{, }c\in\mathbb{R} \)
Miky: \( I=x^5+(3x-3)\cdot x^2+c\text{, }c\in\mathbb{R} \)
1003018708
Level:
A
Given the function \( F(x)= 5\,\mathrm{e}^x+2\sqrt x \), find the function \( f \) such that \( F \) is primitive to \( f \) on \( (0;\infty) \).
\( f(x)=5\,\mathrm{e}^x+\frac{\sqrt x}x \)
\( f(x)=5\,\mathrm{e}^x+\sqrt x \)
\( f(x)=\mathrm{e}^x+\frac{\sqrt x}x \)
\( f(x)=5\,\mathrm{e}^x-\frac{\sqrt x}x \)
1003018707
Level:
A
Given the function \( F(x)=\frac23\cos x-\frac{x^2}2\cdot\ln4 \), find the function \( f \) such that \( F \) is primitive to \( f \) on \(\mathbb{R} \).
\( f(x)=-\frac23\sin x-x\ln4 \)
\( f(x)=\frac23\sin x-x\ln4 \)
\( f(x)=\frac23\sin x-2x\ln2 \)
\( f(x)=-\frac23\sin x-2x \)
1003018706
Level:
A
Evaluate the following integral on \( (0;\infty) \).
\[ \int\left(\frac5x-\frac x5\right)\mathrm{d}x \]
\( 5\ln|x|-\frac{x^2}{10}+c\text{, }c\in\mathbb{R} \)
\( 5 x^0+\frac1{10}x^2+c\text{, }c\in\mathbb{R} \)
\( 5\ln x+\frac{x^2}{10}+c\text{, }c\in\mathbb{R} \)
\( \ln|x|+\frac{x^2}5+c\text{, }c\in\mathbb{R} \)
1003018705
Level:
A
Evaluate the following integral on \( (0;\infty) \).
\[ \int\left(3\sqrt{x^7}-\sqrt[5]{x^4}\right)\mathrm{d}x \]
\( \frac23 x^4\sqrt x-\frac59 x\sqrt[5]{x^4}+c\text{, }c\in\mathbb{R} \)
\( \frac32 x^4\sqrt x-\frac95 x \sqrt[5]{x^4}+c\text{, }c\in\mathbb{R} \)
\( \frac29 x^4\sqrt x-\frac59 x\sqrt[5]{x^4}+c\text{, }c\in\mathbb{R} \)
\( \frac23\sqrt[9]{x^2}-\frac59\sqrt[9]{x^5}+c\text{, }c\in\mathbb{R} \)