2010001603
Level:
B
Assuming \(x\in (1;8)\),
simplify the following expression.
\[
2|x - 8|- 2|1 - x|
\]
\(- 4x + 18\)
\( 14\)
\(4x -18\)
\(- 14\)
B
2010001507
Level:
B
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \frac{19\root{3}\of{x^{4}} - 3}
{\root{4}\of{x^{3}}} \, \mathrm{d}x
\]
\(12(x\root{12}\of{x^{7}} -\root{4}\of{x}) + c,\ c\in \mathbb{R}\)
\(\frac{\frac{57}
{7} \root{3}\of{x^{7}}-3x}
{\frac{4}
{7} \root{4}\of{x^{7}}} + c,\ c\in \mathbb{R}\)
\(\frac{361}
{12} \root{12}\of{x^{19}} -\frac{3}
{4}\root{4}\of{x} + c,\ c\in \mathbb{R}\)
2010001506
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int (x^{4} - 5)^{2}\, \mathrm{d}x
\]
\(\frac{x^{9}}
{9} - 2x^{5} + 25x + c,\ c\in \mathbb{R}\)
\(\frac{(x^{4}-5)^{3}}
{3} + c,\ c\in \mathbb{R}\)
\(8x^{9} - 40x^{5} + 25x + c,\ c\in \mathbb{R}\)
2010001505
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int x2^{x}\, \mathrm{d}x
\]
\(\frac{x2^x}{\ln 2} - \frac{2^x}{\ln^2 2} + c,\ c\in \mathbb{R}\)
\(x2^x -2^x + c,\ c\in \mathbb{R}\)
\(\frac{x^22^x}{2\ln 2}+ c,\ c\in \mathbb{R}\)
\( \frac{2^x(x-1)}{\ln 2} + c,\ c\in \mathbb{R}\)
2010001504
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int x\cos x\, \mathrm{d}x
\]
\( x\sin x +\cos x + c,\ c\in \mathbb{R}\)
\( x\sin x -\cos x + c,\ c\in \mathbb{R}\)
\( -x\sin x +\cos x + c,\ c\in \mathbb{R}\)
\(- x\sin x -\cos x + c,\ c\in \mathbb{R}\)
2010001503
Level:
B
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \frac{2x^{4} -x^2}
{x^{3}} \, \text{d}x
\]
\(x^2 -\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{\frac{2}{5}x^5-\frac{x^3}{3}}{\frac{x^4}{4}}
+ c,\ c\in \mathbb{R}\)
\(2-\ln |x| + c,\ c\in \mathbb{R}\)
\(4x^2 -\ln |x| + c,\ c\in \mathbb{R}\)
2010001502
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int (5x^{2} -2)(2-x^{2} )\, \mathrm{d}x
\]
\(-x^5 +4x^3-4x + c,\ c\in \mathbb{R}\)
\( \left(\frac{5}{3}x^{3} -2x \right)\left(2x -\frac{x^3}{3}\right)
+ c,\ c\in \mathbb{R}\)
\(-20x^3 +24x + c,\ c\in \mathbb{R}\)
\(\frac{4x^3}{3}+ c,\ c\in \mathbb{R}\)
2010001501
Level:
B
Find the indefinite integral $\int\frac{4\ln^3x}x\,\mathrm{d}x$ on $(0;\infty)$. Use a substitution $a=\ln x$.
$\ln^4x+c$, $c\in\mathbb{R}$
$12\ln^2x+c$, $c\in\mathbb{R}$
$2\ln^2x+c$, $c\in\mathbb{R}$
$16\ln^4x+c$, $c\in\mathbb{R}$
2010001405
Level:
B
Find the set difference \(A\setminus B\)
for \(A = \{x\in \mathbb{Z}\ \colon \left |x\right | < 3\}\)
and \(B = \{x\in \mathbb{N}\ \colon x \geq 2\}\).
\(\{ - 2;-1;0;1\}\)
\( \emptyset \)
\(\{ 2\}\)
\(\{ - 2;-1;0\}\)
2010001402
Level:
B
Find the set \( (B\setminus A) \cap C \) for \( A=[ -5;0) \), \( B=[ -1;10) \) and \( C=(-2;2 ] \).
\( [ 0;2]\)
\( ( 0;2]\)
\( [ -2;-1)\)
\( [ -5;2]\)