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.
\(57\, \mathrm{cm}^{2}\)
\(28.5\, \mathrm{cm}^{2}\)
\(27\, \mathrm{cm}^{2}\)
\(35\, \mathrm{cm}^{2}\)
\(45\, \mathrm{cm}^{2}\)

9000066001

Level: 
C
Identify the formula for integration by parts.
\(\int u(x)v'(x)\, \mathrm{d}x = u(x)v(x) -\int u'(x)v(x)\, \mathrm{d}x\)
\(\int u(x)v(x)\, \mathrm{d}x = u'(x)v'(x) -\int u'(x)v(x)\, \mathrm{d}x\)
\(\int u'(x)v'(x)\, \mathrm{d}x = u(x)v(x) -\int u'(x)v(x)\, \mathrm{d}x\)
\(\int u(x)v'(x)\, \mathrm{d}x = u(x)v(x) +\int u'(x)v(x)\, \mathrm{d}x\)

9000066004

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

9000066006

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