1003118904
Level:
B
Evaluate the definite integral.
\[ \int\limits_{-\frac{\pi}2}^{\pi}x\cdot\sin x\,\mathrm{d}x \]
\( \pi+1 \)
\( \pi-1 \)
\( \frac{\pi+2}2 \)
\( \frac{\pi}2 \)
B
1003118903
Level:
B
Evaluate the definite integral.
\[ \int\limits_{-0.5}^{0.1}\mathrm{e}^{5x-1}\mathrm{d}x \]
\( 0.115 \)
\( 0.127 \)
\( 0.121 \)
\( 0.205 \)
1003118902
Level:
B
Evaluate the definite integral.
\[ \int\limits_0^{10}\frac{4x}{x^2+1}\mathrm{d}x \]
\( \ln(10\,201) \)
\( \ln(10\,404) \)
\( \ln(202) \)
\( 9 \)
1003118901
Level:
B
Evaluate the definite integral.
\[ \int\limits_{-2}^3 \sqrt{5x+12}\,\mathrm{d}x \]
\( 18.3 \)
\( 19.1 \)
\( 18.1 \)
\( 19.3 \)
1003261904
Level:
B
Given the function
\[ f(x)=\sin x-3\cos x\text{ ,} \]
determine the set of all \( x \), \( x\in\mathbb{R} \), such that \( f''(x)+f(x)=0 \).
\( \mathbb{R} \)
\( \emptyset \)
\( \{k\pi,\ k\in\mathbb{Z}\} \)
\( \left\{(2k+1)\frac{\pi}2,\ k\in\mathbb{Z} \right\} \)
1003261903
Level:
B
Given the function
\[ f(x)=x^3-3x^2+2\text{ ,} \]
determine the set of all \( x \), \( x\in\mathbb{R} \), such that \( f''(x)-f'(x)=3 \).
\( \{1,3\} \)
\( \{-1,-3\} \)
\( \{-\sqrt3,\sqrt3\} \)
\( \{\sqrt3\} \)
\( \emptyset \)
1003261902
Level:
B
Find the second derivative of the function
\[ f(x)=\sin^2 x \]
at the point \( x_0=-\frac{\pi}6 \).
\( 1 \)
\( \frac12 \)
\( -\frac12 \)
\( -1 \)
\( \sqrt3 \)
\( -\frac{\sqrt3}2 \)
1003261901
Level:
B
Find the second derivative of the function
\[ f(x)=\frac{x^2}{1-x} \]
at the point \( x_0=2 \).
\( -2 \)
\( 2 \)
\( -\frac14 \)
\( \frac14 \)
\( -4 \)
\( 4 \)
1003164006
Level:
B
The value of the expression
\[ \frac{x^4-16}{\left(x^2+4\right)\left(x+2\right) }\]
for \( x=2-\sqrt2 \) is equal to:
\( -\sqrt2 \)
\( \sqrt2 \)
\( 2 \)
\( -2 \)
1003108809
Level:
B
We are given the equation
\[ \sum\limits_{n=1}^{\infty} (\sin x)^{2n-2}=2\cdot\,\mathrm{tg}\,x \]
with the unknown \( x \) being a real number. What is the set of all its solutions?
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac14\pi+k\cdot\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac34\pi+k\cdot\pi\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac34\pi+k\cdot\frac{\pi}2\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac18\pi+k\cdot\frac{\pi}2\right\} \)
\( \bigcup\limits_{k\in\mathbb{Z}}\left\{\frac14\pi+k\cdot\frac{\pi}2\right\} \)