Answer
\[ = 1\]
Work Step by Step
\[\begin{gathered}
f\,\left( x \right) = x\sin \,\left( {{x^2}} \right) \hfill \\
f\,\left( x \right) = 0\,\left( {axis - x} \right) \hfill \\
\hfill \\
let\,\,x = 0\,\,\,and\,\,\,x = \sqrt \pi \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
area\,\,\int_0^{\sqrt \pi } {x\sin {x^2}dx} \hfill \\
\hfill \\
or \hfill \\
\hfill \\
= \frac{1}{2}\int_0^{\sqrt \pi } {\sin \,\left( {{x^2}} \right)\,\left( {2x} \right)dx} \hfill \\
\hfill \\
integrate\,\,\,use\,\,\int {\sin u} du = - \cos u + C \hfill \\
\hfill \\
= - \,\,\left[ {\frac{1}{2}\cos \,\left( {{x^2}} \right)} \right]_0^{\sqrt \pi } \hfill \\
\hfill \\
Fundamental\,\,theorem \hfill \\
\hfill \\
= - \,\,\left[ {\frac{1}{2}\cos \,{{\left( {\sqrt \pi } \right)}^2} - \frac{1}{2}\cos \,{0^2}} \right] \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= - \,\,\left[ {\frac{1}{2}\,\left( { - 1} \right) - \frac{1}{2}} \right] \hfill \\
\hfill \\
= 1 \hfill \\
\hfill \\
\end{gathered} \]
