\begin{homework}\
\label{homework:16}

\medskip


\exercise %1.
Find the anti-derivative of each of the following:

\begin{center}
\begin{tabular}{lll}
 (a) $\frac{1}{ x^4}+x\sqrt{x}+5$;
&(b) $(1-t)^9$; 
&(c) $(x^2+1)^2$; \\
 (d) $1/{\root 3 \of {1-5t}}$; 
&(e) $\sin^3x\cos x$;
&(f) $x\sqrt{100-x^2}$;\\ 
 (g) $x^2/\sqrt{1-x^3}$;
&(h) $\sin\left(\frac{2\pi}{ B}(t-C)\right)+D$; 
&(i) $\cos(\frac{t}{ 10}) \sqrt{1-\sin(\frac{t}{ 10})}$; \\
 (j) $\cos(\pi t)\cos\bigl(\sin(\pi t)\bigr)$.&& 
\end{tabular}\\
\end{center}

\smallskip

\exercise %2.
Suppose that the $\cos(\pi t)$ term were absent in the numerator of
Example~\ref{eg:16.4}(b). Find the anti-derivative by substituting $u=\pi t$ and
then using the formula for the derivative of ~cot.

\smallskip

\exercise %3.
Find the formula for $s(t)$ for a falling body if you know $v_0=v(t_0)$ and
$s_0=s(t_0)$ at some time $t_0$ (not necessarily at time $0$).

\smallskip

\exercise %4.
Find the distance function $s(t)$ if $a(t)=1/t^4$ and both velocity and distance
are zero at time $t=1$.

\smallskip

\exercise %5.
Find the distance function $s(t)$ if the acceleration function
is given by the following formula, and if the initial velocity $v_0$
and the initial distance $s_0$ at time $t=0$ are known: 

\begin{center}
\begin{tabular}{ll}
(a) $a(t)=kt$ (where $k$ is a constant);
&
(b) $a(t)=\sqrt{2t+1}$;\\
(c) $a(t)=\sin(5t)$;
& 
(d) $a(t)=\sin(0.1t)/\cos^3(0.1t)$.\\
\end{tabular}\\
\end{center}

\end{homework}

