1. ## Word Problem Integration

Let $x$ be a function of $t$ such that $\frac{dx}{dt}=2\sin^2 t\cos^2 x$. Suppose that $x\left(\frac{\pi}{4}\right)=\arctan \left(\frac{1}{2}+\frac{\pi}{4}\right)$. Then $x(0)=$

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2. $
\begin{gathered}
\frac{{dx}}
{{dt}} = 2\sin ^2 t\cos ^2 x \hfill \\
\hfill \\
\Leftrightarrow \frac{1}
{{\cos ^2 }}dx = 2\sin ^2 tdt \hfill \\
\end{gathered}$

$
\Leftrightarrow \tan x = t - \frac{1}
{2}\sin 2t + C
$

$
x(t) = a\tan \left( {t - \frac{1}
{2}\sin 2t} \right) + K
$

now apply the I.V. .....