1003108007
Level:
A
What is the difference \( I_1-I_2 \), where \( I_1=\int\limits_{-1}^1\left(\frac x2+2\right)\mathrm{d}x \) and \( I_2=\int\limits_1^2\frac2x\mathrm{d}x \)?
\( 4-\ln 4\)
\( \ln 4 \)
\( 4 + \ln 4 \)
\( \ln 4 - 4 \)
Definite integral
1003108006
Level:
A
Evaluate the definite integral. \[ \int\limits_1^2\left(\frac2x-\frac x2+x^2\right)\mathrm{d}x \]
\( \frac{19}{12}+\ln 4 \)
\( \frac{19}{12}+\ln 2 \)
\( \frac{43}{12} +\ln 4 \)
\( \frac{19}{12} \)
1003108005
Level:
A
Evaluate the definite integral. \[ \int\limits_{\frac{\pi}6}^{\frac{\pi}3}\left(\frac3{\cos^2x} -\frac6{\sin^2x}\right)\mathrm{d}x \]
\( -2\sqrt3 \)
\( 2\sqrt3 \)
\( 0 \)
\( 12\sqrt3 \)
1003108004
Level:
A
Evaluate the definite integral. \[ \int\limits_0^1\left(6\sqrt x-5x^4+3\mathrm{e}^x\right)\mathrm{d}x \]
\( 3\mathrm{e} \)
\( 6+3\mathrm{e} \)
\( 0 \)
\( -3\mathrm{e} \)
1003108003
Level:
A
Evaluate the definite integral. \[ \int\limits_{-\frac{\pi}3}^{\frac{\pi}3}(\sin x-\cos x )\mathrm{d}x \]
\( -\sqrt3 \)
\( 1 \)
\( 1-\sqrt3 \)
\( 1+\sqrt3 \)
1003108002
Level:
A
Evaluate the definite integral. \[ \int\limits_0^1\left(\frac53\cdot\sqrt[3]{x^2}-x^3+\ln 3\right)\mathrm{d}x \]
\( \frac34+\ln 3 \)
\( \frac54+\ln 3 \)
\( \frac34 \)
\( \frac54 \)
1003108001
Level:
A
Evaluate the definite integral. \[ \int\limits_1^2\left(x^7-\ln7\cdot7^x+7x\right)\mathrm{d}x \]
\( \frac38 \)
\( -\frac{51}8 \)
\( \frac{51}8 \)
\( \frac{61}8 \)
1003027503
Level:
C
Evaluate the following definite integral.
\[ \int\limits_{\mathrm{e}}^{12}\frac{x+5}{\frac14 x\cdot(x+4)}\mathrm{d}x \]
\(\ln(\mathrm{e}+4)+5\ln12-2\ln4-5 \)
\( 5+2\ln 4+5\ln12-\ln(\mathrm{e}+4) \)
\( \ln\frac{15}4-5+\ln(4+\mathrm{e}) \)
\( \frac{ \ln \frac{12^5}{16} }{ \ln \frac{\mathrm{e}^5}{\mathrm{e}+4}} \)
1003027502
Level:
C
Evaluate the following definite integral.
\[ \int\limits_5^{25}\frac{4x+1}{x^2-x-2}\mathrm{d}x \]
\( \ln\frac{13}3+3\ln\frac{23}3 \)
\( \ln(26\cdot23^3\cdot6\cdot3^3) \)
\( \ln\frac{26}{27}+3\ln23+\ln6 \)
\( \ln(24\cdot27^3)-\ln(4\cdot7^3) \)
1003027501
Level:
C
Evaluate the following definite integral.
\[ \int_{\mathrm{e}}^{\mathrm{e}^3}\frac{10x-14}{x\cdot(x-2)}\mathrm{d}x \]
\( 3\ln\frac{\mathrm{e}^3-2}{\mathrm{e}-2}+14 \)
\( 3\ln\left[\left(\mathrm{e}^3-2\right)\left(\mathrm{e}-2\right)\right]+14 \)
\( \ln\frac{\mathrm{e}^3-2}{\mathrm{e}-2}+14 \)
\( 3\ln\left(\mathrm{e}^3-2\right)+\ln(\mathrm{e}-2)^3+14 \)