integral of e^x * sin(pi/4 + x/2) limit 0 to 2pi

Integral (lim o to 2 pie) e^x.sin([ie/4+x/2) dx

let pie/4+x/2=t

on diff w r to x we get dx = 2dt

new limits are pie/4 and 5pie/4

integral(lim pie/4 to 5pie/4)e^2(t-pie/4) .sint 2 dt

= 2 e^-pie/2 integral (pie/4 to 5pie/4) e^2t sint dt

=2e^-pie/2[e^2t(2sint+cost]/4+1

=2/5. e^-pie/2[e^5pie/3{2 sin5pie/4-cos5pie/4}-e^2pie/4{2sinpie/4-cospie/4}]

=2/5e^-pie/2[cospie/4(e^5pie/2+e^pie/2)-2 sin(e^5pie/2+e^pie/2)]


Thanks & Regards

Rinkoo Gupta

AskIITians Faculty