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
1003108008
Level:
A
Find the value of \( p \) such that \[ \int\limits_{-2}^3\left(3x^2-5x^4+4x+p\right)\mathrm{d}x=-255. \]
\( -5 \)
\( 5 \)
\( 3 \)
No such \( p \) exists.
1003108106
Level:
A
How many times is \( \int\limits_0^{\frac{\pi}2}3\cos x\,\mathrm{d}x \) bigger than \( \int\limits_{\frac{\pi}2}^{\pi}\frac{\sin x}2\,\mathrm{d}x \)?
\( 6 \) times
\( 3 \) times
\( 2 \) times
It is not bigger.
2000000901
Level:
A
Compare definite integrals \(I_1 = \int\limits_0^{\frac{\pi}{4}}\mathrm{tg}\,x \,\mathrm{d}x\) and \(I_2= \int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}}\mathrm{cotg}\, x\,\mathrm{d}x\).
\(I_1=I_2\)
\(I_1 > I_2\)
\(I_1 < I_2\)
These integrals cannot be compared.
2000000902
Level:
A
Compare definite integrals \(I_1 = \int\limits_{-10}^{10}x^4\,\mathrm{d}x\) and \(I_2= \int\limits_{-10}^{10}x^5\,\mathrm{d}x\).
\(I_1 > I_2\)
\(I_1 = I_2\)
\(I_1 < I_2\)
These integrals cannot be compared.
2000000903
Level:
A
Compare definite integrals \(I_1 = \int\limits_0^{1}x^2 \,\mathrm{d}x\) and \(I_2= \int\limits_{0}^{1}(1-x^2)\,\mathrm{d}x\).
\( I_1 < I_2 \)
\(I_1 =I_2\)
\(I_1 >I_2\)
These integrals cannot be compared.
2000000904
Level:
A
Evaluate the definite integral \(\int\limits_0^{1}x^3\,\mathrm{d}x\).
\(\frac{1}{4}\)
\(1\)
\(\frac{1}{3}\)
\(3\)
2000000905
Level:
A
Evaluate the definite integral \(\int\limits_1^{2}x\, \mathrm{d}x\).
\(\frac{3}{2}\)
\(1\)
\(6\)
\(3\)
2000000906
Level:
A
Evaluate the definite integral \(\int\limits_{\pi}^{2\pi}\sin x\,\mathrm{d}x\).
\(-2\)
\(-1\)
\(-\frac{\pi}{2}\)
\(0\)
2010000001
Level:
A
Vypočítejte určitý integrál. \[ \int\limits_1^2\left(5^x \cdot\ln5 -x^5-4x\right)\mathrm{d}x \]
\( \frac{7}{2} \)
\( -\frac{5}{2} \)
\(- \frac{1}{2} \)
\( -\frac{5}{6} \)