Primitive function

1003018701

Level: 
A
Evaluate the following integral on \( (0;\pi) \). \[ \int\left(\pi\,\mathrm{e}^x-\frac3{\sin^2 x}\right)\mathrm{d}x \]
\( \pi\,\mathrm{e}^x+3\,\mathrm{cotg}⁡\,x+c\text{, }c\in\mathbb{R} \)
\( \pi\,\mathrm{e}^x+3\,\mathrm{tg}⁡\,x+c\text{, }c\in\mathbb{R} \)
\( \pi\,\mathrm{e}^x+\mathrm{cotg}⁡\,x+c\text{, }c\in\mathbb{R} \)
\( \pi\,\mathrm{e}^x-3\,\mathrm{cotg}⁡\,x+c\text{, }c\in\mathbb{R} \)

1003018702

Level: 
A
Evaluate the following integral on \( \mathbb{R} \). \[ \int\left(3^2+3x^2+3^x-\mathrm{e}^x+2^{\mathrm{e}}\right)\mathrm{d}x \]
\( 9x+x^3+\frac{3^x}{\ln⁡3} -\mathrm{e}^x+2^{\mathrm{e}} x+c\text{, }c\in\mathbb{R} \)
\( x^3+\frac{3^x}{\ln⁡3} -\mathrm{e}^x+c\text{, }c\in\mathbb{R} \)
\( 9x+3x^3+3^x-\mathrm{e}^x+\frac{2^{\mathrm{e}+1}}{\mathrm{e}+1}+c\text{, }c\in\mathbb{R} \)
\( 9x+6x+\frac{3^x}{\ln⁡3} -\mathrm{e}^x+2^\mathrm{e} x+c\text{, }c\in\mathbb{R} \)

1003018703

Level: 
A
Evaluate the following integral on \( \left(\frac{\pi}2;\pi\right) \). \[ \int\left(7 \cos⁡ x+\frac5{\cos^2⁡x}+\frac3{\sin^2⁡x}\right)\mathrm{d}x \]
\( 7\sin x+5\,\mathrm{tg⁡}\,x-3\,\mathrm{cotg}\,⁡x+c\text{, }c\in\mathbb{R} \)
\( -7\sin x+5\,\mathrm{tg⁡}\,x+3\,\mathrm{cotg}\,x+c\text{, }c\in\mathbb{R} \)
\( -7\sin x-5\,\mathrm{tg}\,x-3\,\mathrm{cotg}⁡\,x+c\text{, }c\in\mathbb{R} \)
\( 7\sin x-5\,\mathrm{tg}\,x+3\,\mathrm{cotg}\,⁡x+c\text{, }c\in\mathbb{R} \)

1003018704

Level: 
A
Evaluate the following integral on \( (0;\infty) \). \[ \int\left(3\sqrt x+4\sqrt[3]x-35\sqrt[4]{x^3}\right)\mathrm{d}x \]
\( 2x\sqrt x+3x\sqrt[3]x-20x\sqrt[4]{x^3}+c\text{, }c\in\mathbb{R} \)
\( \frac92 x\sqrt x+\frac{16}3 x\sqrt[3]x-\frac{245}4 x\sqrt[4]{x^3}+c\text{, }c\in\mathbb{R} \)
\( 2\sqrt x+3\sqrt[3]x-20\sqrt[4]{x^3}+c\text{, }c\in\mathbb{R} \)
\( 2\sqrt[3]x+3\sqrt[4]{x^3}-20\sqrt[7]{x^4}+c\text{, }c\in\mathbb{R} \)

1003018705

Level: 
A
Evaluate the following integral on \( (0;\infty) \). \[ \int\left(3\sqrt{x^7}-\sqrt[5]{x^4}\right)\mathrm{d}x \]
\( \frac23 x^4\sqrt x-\frac59 x\sqrt[5]{x^4}+c\text{, }c\in\mathbb{R} \)
\( \frac32 x^4\sqrt x-\frac95 x \sqrt[5]{x^4}+c\text{, }c\in\mathbb{R} \)
\( \frac29 x^4\sqrt x-\frac59 x\sqrt[5]{x^4}+c\text{, }c\in\mathbb{R} \)
\( \frac23\sqrt[9]{x^2}-\frac59\sqrt[9]{x^5}+c\text{, }c\in\mathbb{R} \)

1003018706

Level: 
A
Evaluate the following integral on \( (0;\infty) \). \[ \int\left(\frac5x-\frac x5\right)\mathrm{d}x \]
\( 5\ln|x|-\frac{x^2}{10}+c\text{, }c\in\mathbb{R} \)
\( 5 x^0+\frac1{10}x^2+c\text{, }c\in\mathbb{R} \)
\( 5\ln x+\frac{x^2}{10}+c\text{, }c\in\mathbb{R} \)
\( \ln|x|+\frac{x^2}5+c\text{, }c\in\mathbb{R} \)

1003018707

Level: 
A
Given the function \( F(x)=\frac23\cos ⁡x-\frac{x^2}2\cdot\ln⁡4 \), find the function \( f \) such that \( F \) is primitive to \( f \) on \(\mathbb{R} \).
\( f(x)=-\frac23\sin ⁡x-x\ln⁡4 \)
\( f(x)=\frac23\sin ⁡x-x\ln⁡4 \)
\( f(x)=\frac23\sin ⁡x-2x\ln⁡2 \)
\( f(x)=-\frac23\sin ⁡x-2x \)

1003018708

Level: 
A
Given the function \( F(x)= 5\,\mathrm{e}^x+2\sqrt x \), find the function \( f \) such that \( F \) is primitive to \( f \) on \( (0;\infty) \).
\( f(x)=5\,\mathrm{e}^x+\frac{\sqrt x}x \)
\( f(x)=5\,\mathrm{e}^x+\sqrt x \)
\( f(x)=\mathrm{e}^x+\frac{\sqrt x}x \)
\( f(x)=5\,\mathrm{e}^x-\frac{\sqrt x}x \)

1003018709

Level: 
A
Four children evaluated the following integral \( I \) on \( \mathbb{R} \). Who made a mistake? \[ I =\int\left(5x^4+9x^2-6x\right)\mathrm{d}x \]
Paul: \( I =\left(x^3+3\right)\cdot x^2-3x^3+c\text{, }c\in\mathbb{R} \)
Jane: \( I =x^2\left(x^3+3x-3\right)+c\text{, }c\in\mathbb{R} \)
Ann: \( I =x^5+3x^3-3x^2+c\text{, }c\in\mathbb{R} \)
Miky: \( I=x^5+(3x-3)\cdot x^2+c\text{, }c\in\mathbb{R} \)

1003018710

Level: 
A
Four children evaluated the following integral \( I \) on \( (0;\infty) \). Who made a mistake? \[ I =\int\left(\frac18\sqrt[8]{x^3}+\frac12\sqrt{x^9}-\frac15\sqrt[5]{x^6} \right)\mathrm{d}x \]
Paul: \( I =\frac1{11}\left(x^3\sqrt[8]x+x\sqrt{x^5}-x\sqrt[5]{x^2}\right)+c\text{, }c\in\mathbb{R} \)
Jane: \( I =\frac1{11}\left(x\sqrt[8]{x^3}+x^5\sqrt x-x^2\sqrt[5] x+c\right)\text{, }c\in\mathbb{R} \)
Ann: \( I =\frac1{11}\left(x\sqrt[8]{x^3}+x^5\sqrt x-x^2\sqrt[5]x\right)+c\text{, }c\in\mathbb{R} \)
Miky: \( I =\frac x{11}\sqrt[8]{x^3}+\frac{x^5}{11}\sqrt x-\frac{x^2}{11}\sqrt[5]x+c\text{, }c\in\mathbb{R} \)