4 Find \( \int_{-n}^{0} \mathrm{e}^{x} \tan ^{-1}\left(\mathrm{e}^{x}\right) \mathrm{d} x \) in terms of \( n \). Deduce the exact value of \( \int_{-\infty}^{0} \mathrm{e}^{x} \tan ^{-1}\left(\mathrm{e}^{x}\right) \mathrm{dx} \)

Encuentre \( y^{\prime} \) si \( y^{5}+3 y^{2}-2 x^{4}=-4 \)

Question 3 finding a potential for it. (a) Find the volume of the parallelopiped whose coterminus edges are given by the vectors \( \left.\vec{A}=2 \vec{i}-e^{x} \cos y+y z\right) \vec{i}+\left(x z-e^{x} \sin y\right) \vec{j}+(x y+z) \vec{k} \) is conservative by \( 3 \vec{j}+4 \vec{k}, \vec{B}=\vec{i}+2 \vec{j}-\vec{k} \) and \( \vec{C}=3 \vec{i}-\vec{j}+2 \vec{k} \).

ón \( f_{x}=\frac{1}{2} x^{2}+x \) en el punto de abscisa \( x=a \), PASA \( a=4 \)

\( y ^ { \prime \prime \prime } + y = e ^ { x } + e ^ { - x } \cos ( 3 x ) + x ^ { 3 } \)

(b) i. If \( F \) is a conservative field, prove that curl \( \vec{F}=\nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational). ii. Conversely, if \( \nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational), prove that \( \vec{F} \) is conservative. (c) If \( \phi=2 x y z^{2}, \quad \vec{F}=x y \vec{i}-z \vec{j}+x^{2} \vec{k} \) and C is the curve \( x=t^{2}, \quad y=2 t, \quad z=t^{3} \) from \( t=0 \) to \( t=1 \), evaluate the line integrals: i. \( \int_{C} \phi d \vec{r} \). ii. \( \int_{C} \vec{F} \times d \vec{r} \). "Mathematics Can Smile!" ( 2 mar