2010013808
Level:
C
Evaluate the following definite integral.
\[\int\limits_{-1}^{0} \sqrt[4]{x^4}\, \mathrm{d}x\]
\(\frac12\)
\(0\)
\(-\frac12\)
This integral cannot be evaluated.
Definite integral
2010013807
Level:
C
Evaluate the following definite integral.
\[\int\limits_{-2}^{-1}\sqrt{x^2}\, \mathrm{d}x\]
\(\frac32\)
\(0\)
\(-\frac32\)
This integral cannot be evaluated.
2010013806
Level:
C
Let \(\mathrm{sgn}(x) = \begin{cases}
1, & x > 0 \\
0, & x = 0.\\
-1, & x < 0\end{cases}\) Evaluate the following definite integral.
\[\int\limits_{-3}^{-2}\left(\mathrm{sgn}(x+1)-1\right)\, \mathrm{d}x \]
\(-2\)
\(0\)
\(-1\)
This integral cannot be evaluated.
2010013805
Level:
C
Let \(\mathrm{sgn}(x) = \begin{cases}
1, & x > 0 \\
0, & x = 0.\\
-1, & x < 0\end{cases}\) Evaluate the following definite integral.
\[\int\limits_{-2}^{-1}\left(\mathrm{sgn}(x-1)+1\right)\, \mathrm{d}x \]
\(0\)
\(-1\)
\(2\)
This integral cannot be evaluated.
2010013804
Level:
C
Any positive real number \(x\) can be written as \(x=c+d\), where \(c\) is an integer and \(d\in[ 0,1 )\). Then \(c\) is called the integer part of \(x\) and is denoted by \(\left[x\right]\). Evaluate the following definite integral.
\[\int\limits_{3.1}^{\frac72}\left[x\right]\mathrm{d}x \]
\(1.2\)
\(1.6\)
\(3\)
This integral cannot be evaluated.
2010013803
Level:
C
Any positive real number \(x\) can be written as \(x=c+d\), where \(c\) is an integer and \(d\in[ \left. 0,1\right)\). Then \(c\) is called the integer part of \(x\) and is denoted by \(\left[x\right]\). Evaluate the following definite integral.
\[\int\limits_{\frac52}^{2.8}\left[x\right]\,\mathrm{d}x \]
\(0.6\)
\(0.9\)
\(2\)
This integral cannot be evaluated.
2010013802
Level:
C
Evaluate the following definite integral.
\[ \int\limits_{-2}^0\left(1-|x+1|\right)\mathrm{d}x \]
\(1\)
\(2\)
\(-2\)
This integral cannot be evaluated.
2010013801
Level:
C
Evaluate the following definite integral.
\[ \int\limits_0^2\left(|x-1|+1\right)\mathrm{d}x \]
\(3\)
\(2\)
\(-2\)
This integral cannot be evaluated.
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.
2010008005
Level:
A
Compare the two definite integrals \( I_1 = \int_1^2 \left( x^2-x\right) \mathrm{d}x\) and \( I_2 = \int_2^1 \left( x-x^2\right) \mathrm{d}x\).
\( I_1 =I_2\)
\( I_1 > I_2\)
\( I_1 < I_2 \)
These integrals cannot be compared.