[tex]\bf \int\limits_{0}^{\frac{1}{2}}xcos(\underline{\pi x})\cdot dx\\
-----------------------------\\
u=\underline{\pi x}\implies \frac{du}{dx}=\pi \implies \frac{du}{\pi }=dx\\
-----------------------------\\
thus\implies \int\limits_{0}^{\frac{1}{2}}xcos(u)\cdot \cfrac{du}{\pi }
\\\\
\textit{wait a sec, we need the integrand in u-terms}\\
\textit{what the dickens is up with that "x"?}\qquad well\\
-----------------------------\\
u=\pi x\implies \frac{u}{\pi }=x\\
-----------------------------\\[/tex][tex]\bf thus
\\\\
\int\limits_{0}^{\frac{1}{2}}\cfrac{u}{\pi }cos(u)\cdot \cfrac{du}{\pi }\implies
\int\limits_{0}^{\frac{1}{2}}\cfrac{u}{\pi }\cdot \cfrac{cos(u)}{\pi }\cdot du\implies
\cfrac{u}{\pi^2}\int\limits_{0}^{\frac{1}{2}}cos(u)\cdot du\\
-----------------------------\\
\textit{now, let us do the bounds}
\\\\
u\left( \frac{1}{2} \right)=\pi\left( \frac{1}{2} \right)\to \frac{\pi }{2}
\\\\
u\left( 0 \right)=\pi (0)\to 0\\
-----------------------------\\[/tex][tex]\bf thus
\\\\
\cfrac{u}{\pi^2}\int\limits_{0}^{\frac{\pi }{2}}cos(u)\cdot du\implies \cfrac{u}{\pi^2}\cdot sin(u)\implies \left[ \cfrac{u\cdot sin(u)}{\pi^2} \right]_{0}^{\frac{\pi }{2}}
[/tex]
