7.1 Integration By Parts/36

$\int _{0}^{\pi }e^{cost}(sin2t)dt$ = $\int _{0}^{\pi }e^{cost}(2sintcost)dt$ = ${-2}\int _{cos0}^{cos\pi }e^{u}{u}du$ = ${-2}{u}e^{u}{\bigg |}_{cos0}^{cos\pi }-(-2)\int _{cos0}^{cos\pi }e^{u}du$

= ${-2}{u}e^{u}{\bigg |}_{cos0}^{cos\pi }+{2}e^{u}{\bigg |}_{cos0}^{cos\pi }du$ = ${2}{u}e^{u}{\bigg |}_{cos\pi }^{cos0}-{2}e^{u}{\bigg |}_{cos\pi }^{cos0}du$ = ${2}{cos(0)}e^{cos(0)}-{2}{cos(\pi )}e^{cos(\pi )}-{2}e^{cos(0)}+{2}e^{cos(\pi )}$

= ${2}(1)e^{1}-{2}(-1)e^{-1}-{2}e^{1}+{2}e^{-1}$ = ${2}e^{1}+{2}e^{-1}-{2}e^{1}+{2}e^{-1}$ = ${2}e^{-1}+{2}e^{-1}$ = ${4}e^{-1}$

${u}$ = ${cost}$

${du}$ = ${-sint}dt$

${-du}$ = ${sint}dt$

${u}$ = ${u}$ , ${dv}$ = $e^{u}du$

${du}$ = ${du}$ , ${v}$ = $e^{u}$