Primitive function

1003027304

Level: 
A
Choose a pair of functions, \( f_1 \) and \( f_2 \), that are antiderivatives of the same function on \( \mathbb{R} \).
\( f_1(x) = 3+\sin x\text{, }f_2(x)=\cos\left(\frac32\pi+x\right) \)
\( f_1(x) = 5+\sin x\text{, }f_2(x)=-\cos x \)
\( f_1(x) = \sin(x+\pi)\text{, }f_2(x)=\sin x \)
\( f_1(x) = \cos x\text{, }f_2(x)=-\cos x \)

1003107805

Level: 
A
Define the function $f(x)$ so that it holds: $f'(x)=x^5-\sqrt[4]x$ on $(0;\infty)$, $f(1)=-1$.
$f(x)=\frac{x^6}6-\frac45x\sqrt[4]x-\frac{11}{30}$
$f(x)=\frac{x^6}6-\frac45\sqrt[4]{x^5}+\frac{11}{30}$
$f(x)=\frac{x^6}6-\frac54x\sqrt[4]x-\frac{11}{30}$
$f(x)=\frac{x^6}6-\frac54x\sqrt[4]x+\frac{11}{30}$

1003107806

Level: 
A
Define the function $f(x)$ so that the following holds: $f''(x)=\mathrm{e}^x+x^5$ on $\mathbb{R}$, $f(0)=1$, and $f(1)=\frac{43}{42}$.
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+(1-\mathrm{e})x$
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+(-\mathrm{e}-1)x$
$f(x)=\mathrm{e}^x+\frac{7}{6}x^7+x-\mathrm{e}x$
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+\frac{43}{42}$

1003107807

Level: 
A
Find a function $F(x)$, that is primitive to the function $f(x)=2^x\cdot\ln⁡2+4^x\cdot2\ln⁡2+8^x\cdot3\ln⁡2$ on $\mathbb{R}$, and satisfies the condition $F(0)=5$.
$F(x)=2^x+4^x+8^x+2$
$F(x)=\frac{2^x}{\ln 2}+\frac{4^x}{\ln 4}+\frac{8^x}{\ln 8}+2x$
$F(x)=2^x+4^x+8^x+5$
$F(x)=2^x\cdot\ln2+2^{x+1}\cdot\ln2+2^{x+3}\cdot\ln2+5$

2010005101

Level: 
A
Evaluate the following integral on \( \mathbb{R} \). \[ \int\left(2^3+2x^3+\mathrm{e}^x-2^x-2^{\mathrm{e}}\right)\mathrm{d}x \]
\( 8x-0.5x^4+\mathrm{e}^x-\frac{2^x}{\ln⁡2} -2^{\mathrm{e}} x+c,\ c\in\mathbb{R} \)
\( -0.5x^4+\mathrm{e}^x-\frac{2^x}{\ln⁡2} +c,\ c\in\mathbb{R} \)
\( 8x-2x^4+\mathrm{e}^x-2^x-\frac{2^{\mathrm{e}+1}}{\mathrm{e}+1}+c,\ c\in\mathbb{R} \)
\( 4-6x^4+\mathrm{e}^x -\frac{2^x}{\ln⁡2} -2^\mathrm{e} x+c,\ c\in\mathbb{R} \)