Primitive function

2010001507

Level: 
B
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \frac{19\root{3}\of{x^{4}} - 3} {\root{4}\of{x^{3}}} \, \mathrm{d}x \]
\(12(x\root{12}\of{x^{7}} -\root{4}\of{x}) + c,\ c\in \mathbb{R}\)
\(\frac{\frac{57} {7} \root{3}\of{x^{7}}-3x} {\frac{4} {7} \root{4}\of{x^{7}}} + c,\ c\in \mathbb{R}\)
\(\frac{361} {12} \root{12}\of{x^{19}} -\frac{3} {4}\root{4}\of{x} + c,\ c\in \mathbb{R}\)

2010008101

Level: 
B
Evaluate the following integral on the interval \( (0;+\infty)\). \[ \int \left( \frac{1}{\sqrt{x}}+\frac{2}{x} + \frac{3}{x^2} \right) \mathrm{d}x \]
\( 2\sqrt{x} +2 \ln x -\frac{3}{x} +c;~c \in \mathbb{R}\)
\( \frac{3}{2\sqrt{x^3}}+\frac{4}{x^2} + \frac{9}{x^3}+c;~c \in \mathbb{R}\)
\( \frac{2}{3\sqrt{x^3}}+\frac{1}{x^2} + \frac{1}{x^3}+c;~c \in \mathbb{R}\)
\( \frac{\sqrt{x}}{2} +2\ln x - \frac{3}{x} +c;~c \in \mathbb{R}\)

2010008102

Level: 
B
Evaluate the following integral on the interval \( (0;+\infty)\). \[ \int \left( \frac{1}{x} - \frac{2}{x^2} +\frac{3}{\sqrt{x}}\right) \mathrm{d}x \]
\( \ln x +\frac{2}{x}+6\sqrt{x}+c;~c \in \mathbb{R}\)
\( \frac{2}{x^2}-\frac{6}{x^3}+\frac{9}{2\sqrt{x^3}}+c;~c \in \mathbb{R}\)
\( \frac{1}{2x^2}-\frac{2}{3x^3}+\frac{2}{\sqrt{x^3}}+c;~c \in \mathbb{R}\)
\( \ln x +\frac{2}{x}+\frac{3\sqrt{x}}{2}+c;~c \in \mathbb{R}\)

2010008103

Level: 
B
Evaluate the following integral on \( \mathbb{R}\). \[ \int \left( x^2+2\sin^2 x + 3 \mathrm{e}^{2x}\right) \mathrm{d}x \]
\( \frac{x^3}{3}+x-\sin x \cos x + \frac32 \mathrm{e}^{2x}+c;~c \in \mathbb{R}\)
\( 3x^3+x-\sin x \cos x + 6\mathrm{e}^{2x}+c;~c \in \mathbb{R}\)
\( \frac{x^3}{3}+\frac23\sin^3 x +\mathrm{e}^{3x}+c;~c \in \mathbb{R}\)
\( \frac{x^3}{3}-2\cos^2 x + 3\mathrm{e}^{2x}+c;~c \in \mathbb{R}\)

2010008104

Level: 
B
Evaluate the following integral on \( \mathbb{R}\). \[ \int \left( 3x^3 + \mathrm{e}^{2x}-\cos^2 x\right) \mathrm{d}x \]
\( \frac{3x^4}{4}+\frac{\mathrm{e}^{2x}}{2}-\frac{x}2-\frac12\sin x \cos x+c;~c \in \mathbb{R}\)
\( 12x^4+2\mathrm{e}^{2x}-\frac{x}2-\frac12\sin x \cos x+c;~c \in \mathbb{R}\)
\( \frac{3x^4}{4}+\frac{\mathrm{e}^{3x}}{3}-\frac13\cos^3 x +c;~c \in \mathbb{R}\)
\( \frac{3x^4}{4}+\mathrm{e}^{2x}-\sin^2 x +c;~c \in \mathbb{R}\)

2010008105

Level: 
B
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \left( \frac{1}{x} + \frac{1}{x+1}+\frac{x}{x^2+2}\right) \mathrm{d}x \]
\( \ln x + \ln(x+1) + \frac12 \ln(x^2+2)+c;~c \in \mathbb{R}\)
\( \ln x + \ln(x+1) + \frac12 x^2\ln(x^2+2)+c;~c \in \mathbb{R}\)
\( \frac{2}{x^2} + \frac{1}{\frac12 x^2+x} + \frac{\frac12x^2}{\frac13x^3+2x}+c;~c \in \mathbb{R}\)
\( \ln x + \ln(x+1) + \frac{3x}{2(x^2+6)}+c;~c \in \mathbb{R}\)

2010008106

Level: 
B
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \left( \frac{2}{x} + \frac{1}{x+2}+\frac{x}{x^2+1}\right) \mathrm{d}x \]
\( 2\ln x + \ln(x+2) + \frac12 \ln(x^2+1)+c;~c \in \mathbb{R}\)
\( 2\ln x + \ln(x+2) + \frac12 x^2\ln(x^2+1)+c;~c \in \mathbb{R}\)
\( \frac{4}{x^2} + \frac{1}{\frac12 x^2+2x} + \frac{\frac12x^2}{\frac13x^3+x}+c;~c \in \mathbb{R}\)
\( 2\ln x + \ln(x+2) + \frac{3x}{2(x^2+3)}+c;~c \in \mathbb{R}\)

2010008107

Level: 
B
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \left( x\sqrt{x}+x\cos x - x\mathrm{e}^x\right) \mathrm{d}x \]
\( \frac25x^2\sqrt{x}+x\sin x+\cos x-x\mathrm{e}^x+\mathrm{e}^x+c;~c \in \mathbb{R}\)
\( \frac{x^2}2\left(\frac23x^{\frac32}+\sin x-\mathrm{e}^x\right)+c;~c \in \mathbb{R}\)
\( \frac25x^2\sqrt{x}+x\sin x-\cos x-x\mathrm{e}^x+\mathrm{e}^x+c;~c \in \mathbb{R}\)
\( \frac25x^2\sqrt{x}+x\sin x + \cos x-x\mathrm{e}^x-\mathrm{e}^x+c;~c \in \mathbb{R}\)

2010008108

Level: 
B
Evaluate the following integral on the interval \((0;+\infty)\). \[ \int \left( x\sqrt[3]{x}+x\sin x + x\mathrm{e}^x\right) \mathrm{d}x \]
\( \frac37x^2\sqrt[3]{x}+x\cos x+\sin x+x\mathrm{e}^x-\mathrm{e}^x+c;~c \in \mathbb{R}\)
\( \frac{x^2}2\left(\frac34x^{\frac43}-\cos x+\mathrm{e}^x\right)+c;~c \in \mathbb{R}\)
\( \frac37x^2\sqrt[3]{x}+x\cos x-\sin x+x\mathrm{e}^x-\mathrm{e}^x+c;~c \in \mathbb{R}\)
\( \frac37x^2\sqrt[3]{x}-x\cos x + \sin x+x\mathrm{e}^x+\mathrm{e}^x+c;~c \in \mathbb{R}\)

2010008109

Level: 
B
Evaluate the following integral on the interval \(\left(\frac{\pi}{4};\frac{3\pi}{4}\right)\). \[ \int \left(\frac{1}{2x}+\sin 2x - \frac{1}{\cos^2 2x}\right) \mathrm{d}x \]
\( \frac12\left(\ln x - \cos 2x- \mathrm{tg}\, 2x\right) +c;~c \in \mathbb{R}\)
\( \frac12\left(\ln(2x) - \cos 2x- \mathrm{tg }\, 2x \right)+c;~c \in \mathbb{R}\)
\( \ln(2x) - \cos 2x- \mathrm{tg }\, 2x +c;~c \in \mathbb{R}\)
\( \ln(2x) + \cos 2x- \mathrm{cotg }\, 2x +c;~c \in \mathbb{R}\)