Question

Evaluate the integral from 0 to 1/2 of xcospixdx

Answer 1

$\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\\ -----------------------------$$\bf thus \\\\ \int\limits_{0}^{\frac{1}{2}}\cfrac{u}{\pi }cos(u)\cdot \cfrac{du}{\pi }\implies \int\limits_{0}^{\frac{1}{2}}\cfrac{u}{\pi }\cdot \cfrac{cos(u)}{\pi }\cdot du\implies \cfrac{u}{\pi^2}\int\limits_{0}^{\frac{1}{2}}cos(u)\cdot du\\ -----------------------------\\ \textit{now, let us do the bounds} \\\\ u\left( \frac{1}{2} \right)=\pi\left( \frac{1}{2} \right)\to \frac{\pi }{2} \\\\ u\left( 0 \right)=\pi (0)\to 0\\ -----------------------------$$\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}}$