2010008006
Level:
A
Compare the two definite integrals \( I_1 = \int_0^1 \left( x^3-x\right) \mathrm{d}x\) and \( I_2 = \int_1^0 \left( x-x^3\right) \mathrm{d}x\).
\( I_1 =I_2\)
\( I_1 > I_2\)
\( I_1 < I_2 \)
These integrals cannot be compared.
Definite integral
9000150401
Level:
A
Evaluate the following definite integral.
\[
\int _{-3}^{1}(x^{2} + 3x)\, \text{d}x
\]
\(-\frac{8}
{3}\)
\(\frac{8}
{3}\)
\(-\frac{64}
{3} \)
\(\frac{64}
{3} \)
9000150402
Level:
A
Evaluate the following definite integral.
\[
\int _{-\frac{\pi }{
2} }^{ \frac{\pi }
{2} }\sin x\, \text{d}x
\]
\(0\)
\(\pi \)
\(2\)
\(1\)
9000150403
Level:
A
Evaluate the following definite integral.
\[
\int _{-2}^{0}\mathrm{e}^{x}\, \text{d}x
\]
\(1 -\frac{1}
{\mathrm{e}^{2}} \)
\(1 + \frac{1}
{\mathrm{e}^{2}} \)
\(\frac{1}
{\mathrm{e}^{2}} \)
\(-\frac{1}
{\mathrm{e}^{2}} \)
9000150404
Level:
A
Evaluate the following definite integral.
\[
\int _{2}^{6} \frac{2}
{x}\, \text{d}x
\]
\(\ln 9\)
\(\ln 2\)
\(\ln 3\)
\(2\ln 6\)
9000150407
Level:
A
Evaluate the following definite integral.
\[
\int _{1}^{2}7^{x}\, \text{d}x
\]
\(\frac{42}
{\ln 7} \)
\(49\ln 7\)
\(42\)
\(42\ln 7\)
9000150408
Level:
A
Evaluate the following definite integral.
\[
\int _{0}^{ \frac{\pi }{
4} } \frac{2}
{\cos ^{2}x}\, \text{d}x
\]
\(2\)
\(0\)
\(4\)
\(\pi \)
1003027601
Level:
B
How much bigger is \( \int\limits_0^3\left[(x-3)^2+1\right]\!\mathrm{d}x \) than \( \int\limits_3^6 \log_55\,\mathrm{d}x \)?
\( 9 \)
\( 15 \)
\( 4 \)
none of the options
1003027602
Level:
B
How much smaller is \( \int\limits_1^{10}\frac{x+1}{x^2+x}\,\mathrm{d}x \) than \( \int\limits_1^{10}\frac{11-x}{10}\,\mathrm{d}x \)?
more than \( 1 \)
less than \( 1 \)
they are equal
cannot be determined
1003027603
Level:
B
How many times is \( \int\limits_{\frac{\pi}6}^{\frac{\pi}3} \frac{\sin 2x}{\cos x}\,\mathrm{d}x \) bigger than \( \int\limits_{-\frac{\pi}3}^{-\frac{\pi}6} \frac{\sin 2x}{\sin x}\,\mathrm{d}x \)?
they are equal
\( 2 \) times
\( 4 \) times
none of the options