B

9000070707

Level: 
B
Differentiate the following function. \[ f(x) = \root{5}\of{x^{2} - 7x} \]Remark: The function \(f\colon y = \root{5}\of{x}\) is defined for \(x\in \left < 0;\infty \right )\).
\(f^{\prime}(x) = \frac{2x-7} {5(x^{2}-7x)^{\frac{4} {5} }} ;\ x\in \left (-\infty ;0\right )\cup \left (7;\infty \right )\)
\(f^{\prime}(x) = \frac{2x-7} {5(x^{2}-7x)^{\frac{4} {5} }} ;\ x\in \left (-\infty ;0\right ] \cup \left [ 7;\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x};\ x\in \left (-\infty ;0\right )\cup \left (7;\infty \right )\)
\(f^{\prime}(x) = (2x - 7)\root{4}\of{x^{2} - 7x};\ x\in \left (-\infty ;0\right ] \cup \left [ 7;\infty \right )\)

9000066008

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

9000065901

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

9000065903

Level: 
B
Evaluate the following integral on the interval \((-6;+\infty)\). \[ \int \frac{1} {6x + 36}\, \text{d}x \]
\(\frac{1} {6}\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(-\frac{1} {2}(6x + 36)^{-2} + c,\ c\in \mathbb{R}\)
\(6\ln |x + 6| + c,\ c\in \mathbb{R}\)
\(12x^{2} + 36x + c,\ c\in \mathbb{R}\)

9000070110

Level: 
B
Given \(z_{1} = 4\left (\cos \frac{5} {3}\pi + \mathrm{i}\sin \frac{5} {3}\pi \right )\) and \(z_{2} = 2\left (\cos \frac{1} {6}\pi + \mathrm{i}\sin \frac{1} {6}\pi \right )\), evaluate \(\frac{z_{1}} {z_{2}} \).
\(- 2\mathrm{i}\)
\(4\mathrm{i}\)
\(\mathrm{i}\)
\(-\frac{1} {2}\mathrm{i}\)

9000065905

Level: 
B
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \frac{\left (\sqrt{x} + 2\right )^{2}} {x} \, \text{d}x \]
\(x + 8\sqrt{x} + 4\ln |x| + c,\ c\in \mathbb{R}\)
\(\sqrt{x} + 8x + 4\ln |x| + c,\ c\in \mathbb{R}\)
\(\frac{1} {2}x^{-\frac{1} {2} } + 2x +\ln |x| + c,\ c\in \mathbb{R}\)
\(1 + 8\sqrt{x} + 4\ln |x| + c,\ c\in \mathbb{R}\)