1003124306
Level:
C
Which of the positive values of a real number \(a\) makes the equality \( \int\limits_1^a \frac3{3x+5}\,\mathrm{d}x=\ln\frac74 \) true?
\( a = 3 \)
\( a = 2 \)
\( a = 4 \)
\( a = 5 \)
Definite integral
1003124307
Level:
C
Which of the positive values of a real number \(a\) makes the equality \(\int\limits_a^7 \frac{6x-5}{3x^2-5x}\,\mathrm{d}x=\ln 56 \) true?
\( a=2 \)
\( a=1 \)
\( a=3 \)
\( a=5 \)
1003124308
Level:
C
Which of the values of a real number \( a\in(\pi;2\pi) \) makes the equality \( \int\limits_{\pi}^{a+\pi} \sin2x\,\mathrm{d}x=1 \) true?
\( a=\frac32\pi \)
\( a=\frac34\pi \)
\( a=\frac12\pi \)
\( a=\frac23\pi \)
1103124301
Level:
C
The picture shows graphs of two quadratic functions \( f_1(x) \) and \( f_2(x) \). Find the unknown real positive constant \( a \) (as shown in the picture) such that the value of the definite integral \( \int\limits_{-1}^1 f_1(x)\,\mathrm{d}x \) is greater by \( 8 \) than the value of the definite integral \( \int\limits_{-1}^1 f_2(x)\,\mathrm{d}x \).
\( a = 3 \)
\( a = 1 \)
\( a = 4 \)
\( a = 6 \)
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.
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.
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.
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.
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.
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.