Primitive function

9000150104

Level: 
C
Evaluate the following integral on \(\mathbb{R}\). \[ \int \cos x\cdot \left (-3 +\sin x\right )^{5}\, \mathrm{d}x \]
\(\frac{\left (-3+\sin x\right )^{6}} {6} + c\text{, }c\in \mathbb{R}\)
\(6\left (-3 +\sin x\right )^{6} + c,\ c\in \mathbb{R}\)
\(\frac{\left (-3+\cos x\right )^{6}} {6} + c,\ c\in \mathbb{R}\)
\(6\left (-3 +\cos x\right )^{6} + c,\ c\in \mathbb{R}\)

9000150103

Level: 
A
Evaluate the following integral on the interval \(\left(-\frac{\pi}2;\frac{\pi}2\right)\). \[ \int \left ( \frac{3} {\cos ^{2}x} - 3\mathrm{e}^{x}\right )\, \mathrm{d}x \]
\(3\mathop{\mathrm{tg}}\nolimits x - 3\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(- 3\mathop{\mathrm{tg}}\nolimits x - 3\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(- 3\mathop{\mathrm{tg}}\nolimits x + 3\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)
\(3\mathop{\mathrm{tg}}\nolimits x + 3\mathrm{e}^{x} + c,\ c\in \mathbb{R}\)

9000071204

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

9000071205

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

9000071208

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

9000071203

Level: 
B
Evaluate the following integral on the interval \((0;\frac{\pi}2)\). \[ \int \frac{\cos 2x} {\sin ^{2}x}\, \mathrm{d}x \]
\(- 2x -\mathop{\mathrm{cotg}}\nolimits x + c,\ c\in \mathbb{R}\)
\(\frac{\sin 2x} {-\frac{1} {3} \cos ^{3}x} + c,\ c\in \mathbb{R}\)
\(\mathop{\mathrm{tg}}\nolimits x - 2x + c,\ c\in \mathbb{R}\)

9000071207

Level: 
B
Evaluate the following integral on the interval \(\left(\sqrt{\frac43};+\infty\right)\). \[ \int \frac{6x} {(3x^{2} - 4)^{2}}\, \mathrm{d}x \]
\(\frac{1} {4-3x^{2}} + c,\ c\in \mathbb{R}\)
\(\frac{3x^{2}} {x^{3}-12x^{2}+16x} + c,\ c\in \mathbb{R}\)
\(\frac{1} {(3x^{2}-4)^{2}} + c,\ c\in \mathbb{R}\)