A

9000065902

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

9000070102

Level: 
A
Evaluate the following complex number. \[ \left (\cos \frac{\pi } {3} + \mathrm{i}\sin \frac{\pi } {3}\right )^{10} \]
\(-\frac{1} {2} -\mathrm{i}\frac{\sqrt{3}} {2} \)
\(-\frac{\sqrt{3}} {2} -\frac{1} {2}\mathrm{i}\)
\(-\frac{\sqrt{3}} {2} + \frac{1} {2}\mathrm{i}\)
\(-\frac{1} {2} + \mathrm{i}\frac{\sqrt{3}} {2} \)

9000065908

Level: 
A
Given the function \[ F(x) = \frac{1} {2}x^{2} - x, \] find the function \(f\) such that \(F\) is primitive to \(f\) on \((1,+\infty )\).
\(f(x) = \frac{x^{2}-1} {x+1} \)
\(f(x) = \frac{x^{2}-1} {x-1} \)
\(f(x) = \frac{x+1} {x^{2}-1}\)
\(f(x) = \frac{x-1} {x^{2}-1}\)

9000065910

Level: 
A
Given function \[ F(x) = x + 2\ln |x|-\frac{1} {x}, \] find the function \(f\) such that \(F\) is primitive to \(f\) on \((0,+\infty )\).
\(f(x) = \frac{x^{2}+2x+1} {x^{2}} \)
\(f(x) = \frac{x^{2}} {(x+1)^{2}} \)
\(f(x) = \frac{x^{2}-1} {x^{2}} \)
\(f(x) = \frac{x^{2}} {(x-1)^{2}} \)