Primitive function

2010005102

Level: 
A
Evaluate the following integral on \( \left(\frac{\pi}2;\pi\right) \). \[ \int\left(5 \sin x-\frac3{\cos^2⁡x}-\frac7{\sin^2⁡x}\right)\mathrm{d}x \]
\( -5\cos x-3\,\mathrm{tg⁡}\,x+7\,\mathrm{cotg}\,⁡x+c,\ c\in\mathbb{R} \)
\( 5\cos x+3\,\mathrm{tg⁡}\,x+7\,\mathrm{cotg}\,x+c,\ c\in\mathbb{R} \)
\( -5\cos x-3\,\mathrm{tg}\,x-7\,\mathrm{cotg}⁡\,x+c,\ c\in\mathbb{R} \)
\( 5\cos x+3\,\mathrm{tg}\,x+7\,\mathrm{cotg}\,⁡x+c,\ c\in\mathbb{R} \)

2010005103

Level: 
A
Evaluate the following integral on \( (0;\infty) \). \[ \int\left(6\sqrt x-5\sqrt[3]{x^2}+10\sqrt[4]{x}\right)\mathrm{d}x \]
\( 4x\sqrt x-3x\sqrt[3]{x^2}+8x\sqrt[4]{x}+c,\ c\in\mathbb{R} \)
\( 9 x\sqrt x-\frac{25}3 x^3\sqrt[3]{x^2}+\frac{25}2 x\sqrt[4]{x}+c,\ c\in\mathbb{R} \)
\( 4\sqrt x-3\sqrt[3]{x^2}+8\sqrt[4]{x}+c,\ c\in\mathbb{R} \)
\(x+20\sqrt{x}+c,\ c\in\mathbb{R} \)

2010005104

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

9000065501

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

9000065503

Level: 
A
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int (4x^{-3} - x^{-4})\, \mathrm{d}x \]
\(- 2x^{-2} + \frac{1} {3}x^{-3} + c,\ c\in \mathbb{R}\)
\(-\frac{4} {3}x^{-2} -\frac{1} {3}x^{-3} + c,\ c\in \mathbb{R}\)
\(-\frac{3} {4}x^{-4} -\frac{1} {5}x^{-5} + c,\ c\in \mathbb{R}\)
\(- 12x^{2} + 4x^{-3} + c,\ c\in \mathbb{R}\)