C

9000071208

Level: 
C
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int x^{2}\ln x\, \mathrm{d}x \]
\(\frac{x^{3}} {3} \left (\ln x -\frac{1} {3}\right ) + c,\ c\in \mathbb{R}\)
\(\frac{x^{2}} {3} + c,\ c\in \mathbb{R}\)
\(x^{2}\left (\frac{x\ln x} {3} -\frac{1} {2}\right ) + c,\ c\in \mathbb{R}\)

9000070505

Level: 
C
The dimensions of the box form a geometric sequence. The volume of the box is \(27\, \mathrm{cm}^{3}\) and the length of the shortest side is \(2\, \mathrm{cm}\). Find the surface area of the box.
\(57\, \mathrm{cm}^{2}\)
\(28.5\, \mathrm{cm}^{2}\)
\(27\, \mathrm{cm}^{2}\)
\(35\, \mathrm{cm}^{2}\)
\(45\, \mathrm{cm}^{2}\)

9000066010

Level: 
C
Evaluate the following integral on \(\mathbb{R}\). \[ \int \mathrm{e}^{2x}\, \mathrm{d}x \]
\(\frac{1} {2}\mathrm{e}^{2x} + c,\ c\in \mathbb{R}\)
\(\frac{1} {3}\mathrm{e}^{3x} + c,\ c\in \mathbb{R}\)
\(\mathrm{e}^{2x} -\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(2\mathrm{e}^{2x} + c,\ c\in \mathbb{R}\)

9000066007

Level: 
C
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int x^{2}\ln x\, \mathrm{d}x \]
\(\frac{1} {3}x^{3}\ln x -\frac{1} {9}x^{3} + c,\ c\in \mathbb{R}\)
\(\frac{1} {2}x^{2}\ln x -\frac{1} {4}x^{2} + c,\ c\in \mathbb{R}\)
\(x\ln x -\frac{1} {2}x^{2} + c,\ c\in \mathbb{R}\)
\(x\ln x - x + c,\ c\in \mathbb{R}\)