Primitive function

9000071202

Level: 
B
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \frac{11\sqrt{x^{3}} - 2} {\root{3}\of{x^{2}}} \, \mathrm{d}x \]
\(6(x\root{6}\of{x^{5}} -\root{3}\of{x}) + c,\ c\in \mathbb{R}\)
\(\frac{\frac{22} {5} \sqrt{x^{5}}-2x} {\frac{3} {5} \root{3}\of{x^{5}}} + c,\ c\in \mathbb{R}\)
\(\frac{121} {6} \root{6}\of{x^{11}} -\frac{2} {3}\root{3}\of{x} + c,\ c\in \mathbb{R}\)

9000065903

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

9000065905

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

9000066008

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

9000065906

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

9000065907

Level: 
B
Evaluate the following integral on \(\mathbb{R}\). \[ \int \frac{x^{4} - 1} {x^{2} + 1}\, \text{d}x \]
\(\frac{1} {3}x^{3} - x + c,\ c\in \mathbb{R}\)
\(\frac{1} {3}x^{3} + x + c,\ c\in \mathbb{R}\)
\(\frac{1} {5}x^{5} - x +\ln |x^{2} - 1| + c,\ c\in \mathbb{R}\)
\(3x^{2} -\ln |x^{2} - 1| + c,\ c\in \mathbb{R}\)

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 \((1;+\infty )\).
\(f(x) = \frac{x^{2}-1} {x+1} \)
\(f(x) = \frac{x^{2}-1} {x-1} \)
\(f(x) = \frac{x+1} {x^{2}-1}\)
\(f(x) = \frac{x-1} {x^{2}-1}\)