\( i _ { v } ( x ) = \frac { \sqrt { \pi } ( \frac { 1 } { 2 } x ) ^ { v } } { \Gamma ( v + \frac { 1 } { 2 } ) } \int _ { 1 } ^ { + \infty } ( t ^ { 2 } - 1 ) ^ { v - \frac { 1 } { 2 } } e ^ { - x t } d t \quad v > - \frac { 1 } { 2 } , x > 0 \)

\( \left. \begin{array} { l } { f ^ { \prime } ( x ) = 4 \sqrt { x ^ { 3 } } + k x ^ { - 2 } } \\ { f ^ { \prime \prime } ( x ) = } \end{array} \right. \)

The area of a rectangular cross-section of a solid is given by the dimensions in terms of a linear function: width = \(x\) and height = \(3 - x\) for \(0 \leq x \leq 3\). Calculate the volume of the solid when these rectangles are stacked from \(x=0\) to \(x=3\).

If \( \left\langle a_{n}\right\rangle \) and \( \left\langle b_{n}\right\rangle \) are two sequences such that \( \lim _{n \rightarrow \infty} a_{n}=i, \lim _{n \rightarrow \infty} b_{n}=m \), then prove that: \( \lim _{n \rightarrow \infty}\left(a_{n}+b_{n}\right)=l+m \)

18) \( \sum_{k=1}^{4} \frac{2 k-1}{k} \)

If a car has an acceleration function of \( a(t) = 6t \), what is its velocity function assuming it starts from rest?