1103107803
Level:
B
Which of the following pictures shows the graph of a function $F(x)$, that is primitive to the function $f(x)=\frac{x^2-11x+28}{x-7}$ on $(-\infty;7)$?
Primitive function
2010000301
Level:
B
Evaluate the following integral on the interval \(\left(0;\frac{\pi}2\right)\).
\[
\int \frac{\cos 2x}
{\cos ^{2}x}\, \mathrm{d}x
\]
\(2x -\mathop{\mathrm{tg}}\nolimits x + c,\ c\in \mathbb{R}\)
\(\frac{\sin 2x}
{\frac{1}
{3} \sin ^{3}x} + c,\ c\in \mathbb{R}\)
\(2x +\mathop{\mathrm{cotg}}\nolimits x + c,\ c\in \mathbb{R}\)
2010000302
Level:
B
Evaluate the following integral on the interval \(\left(\sqrt{\frac34};+\infty\right)\).
\[
\int \frac{8x}
{(4x^{2} - 3)^{2}}\, \mathrm{d}x
\]
\(\frac{1}
{3-4x^{2}} + c,\ c\in \mathbb{R}\)
\(\frac{4x^{2}}
{\frac{16}{5}x^{5}-8x^{3}+9x} + c,\ c\in \mathbb{R}\)
\(\frac{1}
{4x^{2}-3} + c,\ c\in \mathbb{R}\)
2010000303
Level:
B
Evaluate the following integral on the interval \(\left(\frac54;+\infty\right)\).
\[
\int \frac{3}
{5 - 4x}\, \mathrm{d}x
\]
\(-\frac{3}
{4}\ln |5 - 4x| + c,\ c\in \mathbb{R}\)
\(-\frac{3}
{4\cdot \ln |5-4x|} + c,\ c\in \mathbb{R}\)
\(\frac{3}
{4}\ln |5 - 4x| + c,\ c\in \mathbb{R}\)
\( \frac{3}
{4\cdot \ln |5-4x|} + c,\ c\in \mathbb{R}\)
2010001501
Level:
B
Find the indefinite integral $\int\frac{4\ln^3x}x\,\mathrm{d}x$ on $(0;\infty)$. Use a substitution $a=\ln x$.
$\ln^4x+c$, $c\in\mathbb{R}$
$12\ln^2x+c$, $c\in\mathbb{R}$
$2\ln^2x+c$, $c\in\mathbb{R}$
$16\ln^4x+c$, $c\in\mathbb{R}$
2010001502
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int (5x^{2} -2)(2-x^{2} )\, \mathrm{d}x
\]
\(-x^5 +4x^3-4x + c,\ c\in \mathbb{R}\)
\( \left(\frac{5}{3}x^{3} -2x \right)\left(2x -\frac{x^3}{3}\right)
+ c,\ c\in \mathbb{R}\)
\(-20x^3 +24x + c,\ c\in \mathbb{R}\)
\(\frac{4x^3}{3}+ c,\ c\in \mathbb{R}\)
2010001503
Level:
B
Evaluate the following integral on the interval \((0;+\infty)\).
\[
\int \frac{2x^{4} -x^2}
{x^{3}} \, \text{d}x
\]
\(x^2 -\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{\frac{2}{5}x^5-\frac{x^3}{3}}{\frac{x^4}{4}}
+ c,\ c\in \mathbb{R}\)
\(2-\ln |x| + c,\ c\in \mathbb{R}\)
\(4x^2 -\ln |x| + c,\ c\in \mathbb{R}\)
2010001504
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int x\cos x\, \mathrm{d}x
\]
\( x\sin x +\cos x + c,\ c\in \mathbb{R}\)
\( x\sin x -\cos x + c,\ c\in \mathbb{R}\)
\( -x\sin x +\cos x + c,\ c\in \mathbb{R}\)
\(- x\sin x -\cos x + c,\ c\in \mathbb{R}\)
2010001505
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int x2^{x}\, \mathrm{d}x
\]
\(\frac{x2^x}{\ln 2} - \frac{2^x}{\ln^2 2} + c,\ c\in \mathbb{R}\)
\(x2^x -2^x + c,\ c\in \mathbb{R}\)
\(\frac{x^22^x}{2\ln 2}+ c,\ c\in \mathbb{R}\)
\( \frac{2^x(x-1)}{\ln 2} + c,\ c\in \mathbb{R}\)
2010001506
Level:
B
Evaluate the following integral on \(\mathbb{R}\).
\[
\int (x^{4} - 5)^{2}\, \mathrm{d}x
\]
\(\frac{x^{9}}
{9} - 2x^{5} + 25x + c,\ c\in \mathbb{R}\)
\(\frac{(x^{4}-5)^{3}}
{3} + c,\ c\in \mathbb{R}\)
\(8x^{9} - 40x^{5} + 25x + c,\ c\in \mathbb{R}\)