9000150406
Level:
B
Evaluate the following definite integral.
\[
\int _{-3}^{2} \frac{x}
{x^{2} + 3}\, \text{d}x
\]
\(\frac{1}
{2}\ln \frac{7}
{12}\)
\(2\ln \frac{12}
{7} \)
\(2\ln \frac{7}
{12}\)
\(\frac{1}
{2}\ln \frac{12}
{7} \)
Definite integral
1003027501
Level:
C
Evaluate the following definite integral.
\[ \int_{\mathrm{e}}^{\mathrm{e}^3}\frac{10x-14}{x\cdot(x-2)}\mathrm{d}x \]
\( 3\ln\frac{\mathrm{e}^3-2}{\mathrm{e}-2}+14 \)
\( 3\ln\left[\left(\mathrm{e}^3-2\right)\left(\mathrm{e}-2\right)\right]+14 \)
\( \ln\frac{\mathrm{e}^3-2}{\mathrm{e}-2}+14 \)
\( 3\ln\left(\mathrm{e}^3-2\right)+\ln(\mathrm{e}-2)^3+14 \)
1003027502
Level:
C
Evaluate the following definite integral.
\[ \int\limits_5^{25}\frac{4x+1}{x^2-x-2}\mathrm{d}x \]
\( \ln\frac{13}3+3\ln\frac{23}3 \)
\( \ln(26\cdot23^3\cdot6\cdot3^3) \)
\( \ln\frac{26}{27}+3\ln23+\ln6 \)
\( \ln(24\cdot27^3)-\ln(4\cdot7^3) \)
1003027503
Level:
C
Evaluate the following definite integral.
\[ \int\limits_{\mathrm{e}}^{12}\frac{x+5}{\frac14 x\cdot(x+4)}\mathrm{d}x \]
\(\ln(\mathrm{e}+4)+5\ln12-2\ln4-5 \)
\( 5+2\ln 4+5\ln12-\ln(\mathrm{e}+4) \)
\( \ln\frac{15}4-5+\ln(4+\mathrm{e}) \)
\( \frac{ \ln \frac{12^5}{16} }{ \ln \frac{\mathrm{e}^5}{\mathrm{e}+4}} \)
1003118905
Level:
C
Evaluate the definite integral.
\[ \int\limits_{\mathrm{e}}^{\mathrm{e}^2}\frac{\ln x}x\,\mathrm{d}x \]
\( 1.5 \)
\( 2.5 \)
\( \mathrm{e}^2 \)
\( \mathrm{e}^2-\mathrm{e} \)
1003118906
Level:
C
Evaluate the definite integral.
\[ \int\limits_{1}^{\mathrm{e}}\left(\sqrt[3]x-x^3\right)\cdot\ln x\,\mathrm{d}x \]
\( -9.03 \)
\( 9.03 \)
\( 11.4 \)
\( -11.4 \)
1003124302
Level:
C
Which of the given values of a real number \( a\in\left(\frac{\pi}2;\pi\right) \) makes the equality \( \int\limits_a^{2a} (3\sin x-4x)\,\mathrm{d}x=-6a^2 \) true?
\( a =\frac23\pi \)
\( a=\frac56\pi \)
\( a=\frac34\pi \)
\( a=\frac78\pi \)
1003124303
Level:
C
Which of the given values of real numbers \( a \), \( b\in\left(0;\frac{\pi}2\right) \), such as \( a < b \), makes the equality \( \int\limits_a^b \cos x\,\mathrm{d}x=2\cos\frac{\pi}4\cdot\sin\frac{\pi}{12} \) true?
\( a=\frac{\pi}6 \), \( b=\frac{\pi}3 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}6 \)
\( a=\frac{\pi}3 \), \( b=\frac{\pi}4 \)
\( a=\frac{\pi}4 \), \( b=\frac{\pi}3 \)
1003124304
Level:
C
Given a function \( f(x)=ax^4+bx \), find real numbers \( a \) and \( b \), such that \( \int\limits_0^1f(x)\,\mathrm{d}x=27 \) and \( \int\limits_{-1}^0f(x)\,\mathrm{d}x=57 \).
\( a=210 \), \( b=-30 \)
\( a=210 \), \( b=30 \)
\( a=75 \), \( b=60 \)
\( a=30 \), \( b=210 \)
1003124305
Level:
C
Given a function \( f(x)=ax^6+bx^3+cx+8 \), find real numbers \( a \), \( b \) and \( c \), such that \( \int\limits_0^1f(x)\,\mathrm{d}x=\frac{35}4 \), \( f'(0)=2 \) and \( f'(1)=180 \).
\( a=7 \), \( b=-5 \), \( c=2 \)
\( a=7 \), \( b=5 \), \( c=2 \)
\( a=-7 \), \( b=-5 \), \( c=2 \)
\( a=-7 \), \( b=5 \), \( c=-2 \)