For the following two (02) items: Let u = ∫ e^x x\, dx and v = ∫...

Given $u = \int e^x \cos x\, dx$ and $v = \int e^x \sin x\, dx$.

Differentiating both sides with respect to $x$, we get:
$\dfrac{du}{dx} = e^x \cos x$
$\dfrac{dv}{dx} = e^x \sin x$

Evaluating the integrals, we have:
$u = \dfrac{e^x}{2}(\cos x + \sin x) + C_1$
$v = \dfrac{e^x}{2}(\sin x - \cos x) + C_2$

Comparing the derivatives with $u$ and $v$, it is clear that:
$\dfrac{du}{dx} \neq -v$
$\dfrac{dv}{dx} \neq -u$

Therefore, neither statement I nor statement II is correct.