Let $f$ be a differentiable function such that $\int f ( x ) \sin x \, d x = - f ( x ) \cos x + \int 4 x ^ { 3 } \cos x \, d x$. Which of the following could be $f ( x )$ ?
(A) $\cos x$
(B) $\sin x$
(C) $4 x ^ { 3 }$
(D) $- x ^ { 4 }$
(E) $x ^ { 4 }$

Let $f$ be the function defined on $] 0 ; + \infty \left[ \text{ by } f ( x ) = x ^ { 2 } \ln x \right.$.
A primitive $F$ of $f$ on $] 0$; $+ \infty [$ is defined by: a. $F ( x ) = \frac { 1 } { 3 } x ^ { 3 } \left( \ln x - \frac { 1 } { 3 } \right)$; b. $F ( x ) = \frac { 1 } { 3 } x ^ { 3 } ( \ln x - 1 )$; c. $F ( x ) = \frac { 1 } { 3 } x ^ { 2 }$; d. $F ( x ) = \frac { 1 } { 3 } x ^ { 2 } ( \ln x - 1 )$.

Let $f$ be the function defined on $\mathbb{R}$ by $f(x) = x \mathrm{e}^{x}$.
A primitive $F$ on $\mathbb{R}$ of the function $f$ is defined by:
A. $F(x) = \frac{x^{2}}{2} \mathrm{e}^{x}$
B. $F(x) = (x - 1) \mathrm{e}^{x}$
C. $F(x) = (x + 1) \mathrm{e}^{x}$
D. $F(x) = \frac{2}{x} \mathrm{e}^{x^{2}}$.

A continuous increasing function $f ( x )$ on the closed interval $[ 0,1 ]$ satisfies $$\int _ { 0 } ^ { 1 } f ( x ) d x = 2 , \quad \int _ { 0 } ^ { 1 } | f ( x ) | d x = 2 \sqrt { 2 }$$ When the function $F ( x )$ is defined as $$F ( x ) = \int _ { 0 } ^ { x } | f ( t ) | d t \quad ( 0 \leq x \leq 1 )$$ what is the value of $\int _ { 0 } ^ { 1 } f ( x ) F ( x ) d x$? [4 points]
(1) $4 - \sqrt { 2 }$
(2) $2 + \sqrt { 2 }$
(3) $5 - \sqrt { 2 }$
(4) $1 + 2 \sqrt { 2 }$
(5) $2 + 2 \sqrt { 2 }$

If $\int f(x)\, dx = \psi(x)$, then $\int x^5 f\left(x^3\right) dx$, is equal to
(1) $\frac{1}{3}x^3 \psi\left(x^3\right) - \int x^2 \psi\left(x^3\right) dx + c$
(2) $\frac{1}{3}\left[x^3 \psi\left(x^3\right) - \int x^3 \psi\left(x^3\right) dx\right] + c$
(3) $\frac{1}{3}\left[x^3 \psi\left(x^3\right) - \int x^2 \psi\left(x^3\right) dx\right] + c$
(4) $\frac{1}{3}x^3 \psi\left(x^3\right) - 3\int x^3 \psi\left(x^3\right) dx + c$