Question

Evaluate the integral from 0 to 1/2 of xcospixdx

Answer 1

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