Volume of a Solid
Definite integral
1003108205
Level:
B
Compare the two definite integrals \( I_1=\int\limits_{-1}^1\left(x+\frac{\pi}2\right)\mathrm{d}x \) and \( I_2=\int\limits_0^{\frac{\pi}4}\mathrm{tg}\,x\cdot\cos x\,\mathrm{d}x \).
\( I_1 \) is bigger than \( I_2 \).
\( I_1 \) is smaller than \( I_2 \).
\( I_1 \) is equal to \( I_2 \).
These integrals cannot be compared.
1003108204
Level:
B
The value of the definite integral \( \int\limits_{-\frac{\pi}4}^{\frac{\pi}4}\left(\mathrm{tg}^2x+1\right)\mathrm{d}x \) is:
a whole number
a decimal number
a proper fraction
an irrational number
1003108203
Level:
B
Compare the definite integral \( I=\int\limits_0^{\frac{\pi}4}\frac{\cos2b}{\cos^2b}\,\mathrm{d}b \) to the number \( \frac{\pi}2 \).
\( I \) is smaller than \( \frac{\pi}2 \) by \( 1 \).
\( I \) is bigger than \( \frac{\pi}2 \) by \( 1 \).
\( I \) is equal to \( \frac{\pi}2 \).
\( I \) is smaller than \( \frac{\pi}2 \) by \( \frac{\pi}4 \).
1003108202
Level:
B
Evaluate the definite integral \( \int\limits_{\frac{\pi}6}^{\frac{\pi}3}\frac{\mathrm{tg}\,a}{\sin2a}\,\mathrm{d}a \).
\( \frac{\sqrt3}3 \)
\( \frac{2\sqrt3}3 \)
\( -\frac{2\sqrt3}3 \)
\( -\frac{\sqrt3}3 \)
1003108201
Level:
B
Evaluate the definite integral \( \int\limits_0^{\frac{\pi}6}\frac{3\cos2t}{\cos t+\sin t}\,\mathrm{d}t \). Which of the following intervals contains the value of the integral?
\( (0.8;1.2) \)
\( (0.4;0.8) \)
\( (-0.8;-0.1) \)
\( (-0.1;0.4) \)
1003108108
Level:
B
Compare the value of \( \int\limits_1^2\frac{x^2-x}{\sqrt x}\,\mathrm{d}x \) to the number \( \frac4{15} \).
It is bigger than \( \frac4{15} \).
It is smaller than \( \frac4{15} \).
It equals \( \frac4{15} \).
It cannot be compared.
1003108107
Level:
B
Compare the value of \( \int\limits_1^2\frac{x^2\cdot\sqrt[3]x}{\sqrt[4]{x^3}}\,\mathrm{d}x \) to zero.
It is bigger than \( 0 \).
It is smaller than \( 0 \).
It equals \( 0 \).
It cannot be compared.
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.
1003108105
Level:
B
Evaluate the integral \( \int\limits_0^1\frac{x-1}{x+3}\,\mathrm{d}x \).
\( 1+\ln\left(\frac34\right)^4 \)
\( 1-4\ln12 \)
\( 4\ln0.75 \)
\( 4\ln12 \)