Primitive function

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 $F(0)=5$.
$F(x)=2^x+4^x+8^x+2$
$F(x)=\frac{2^x}{\ln 2}+\frac{4^x}{\ln 4}+\frac{8^x}{\ln 8}+2x$
$F(x)=2^x+4^x+8^x+5$
$F(x)=2^x\cdot\ln2+2^{x+1}\cdot\ln2+2^{x+3}\cdot\ln2+5$

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 $f(1)=\frac{43}{42}$.
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+(1-\mathrm{e})x$
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+(-\mathrm{e}-1)x$
$f(x)=\mathrm{e}^x+\frac{7}{6}x^7+x-\mathrm{e}x$
$f(x)=\mathrm{e}^x+\frac{x^7}{42}+\frac{43}{42}$

1003107805

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

1003107804

Level: 
B
Four girls evaluated the integral $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x$ on $\mathbb{R}$. Ann started to integrate by parts like this: $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=\sin^2⁡x-\int\cos x\cdot\sin x\,\mathrm{d}x$. Beth started to integrate by parts like this: $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=-\cos^2 x-\int\sin x\cdot\cos x\,\mathrm{d}x$. Claire used substitution $a=\sin ⁡x$ like this: $I=\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=\int a\,\mathrm{d}a$. Diana integrated $\int\sin ⁡x\cdot\cos x\,\mathrm{d}x=-\cos x\cdot\sin⁡ x+c$, $c\in\mathbb{R}$. Which of the girls made a mistake?
Diana
Ann
Beth
Claire

1003107913

Level: 
C
Which method is the most effective to solve the indefinite integral \[ \int\sin(\ln x)\mathrm{d}x \] in the range \( (0;\infty) \)?
By parts integration, when we let \( u(x)=\sin⁡(\ln ⁡x) \), where \( u(x) \) is the integrated function, and we let \( v'(x)=1 \), where \( v'(x) \) is the differentiated function.
By substitution, \( a=\sin ⁡x \).
By parts integration, when we let \( u(x)=\ln x \), where \( u(x) \) is the integrated function, and we let \( v'(x)=\sin x \), where \( v'(x) \) is the differentiated function.
By substitution, \( t=\sin⁡(\ln⁡ x) \).

1003107912

Level: 
C
Which method is the most effective to solve the indefinite integral \[ \int\frac{\mathrm{d}x}{x\ln ⁡x} \] in the range \( (1;\infty) \)?
By substitution, \( a=\ln ⁡x \).
By parts integration, when we let \( u(x)=\frac1x \), where \( u(x) \) is the integrated function, and we let \( v'(x)=\ln ⁡x \), where \( v'(x) \) is the differentiated function.
By substitution, \( a=\frac1x \).
By factorization into \( \int\frac1x\mathrm{d}x\cdot\int\frac1{\ln ⁡x}\mathrm{d}x \).