intigration of (e^(x/2) sin(x/2+pi/4)) in the interval of [0,2pi]

we must use integration by parts. take e^(x/2) as first part and sin(x/2 + pi/2) as second part. interval [0,2pi] changes to 2*(integral 0 to pi/2). here comes the answer.

an easier method is to take x/2 as t. Then dx=2d. The form it takes then will be like -- 2*[integral of (e^t(sin(t+pi/4))). Now expand sin(t+pi/4) as sinAcosB+cosAsinB. sin pi/4=cos pi/4= 1/root 2. Take 1/root 2 out of the integral. Now you have the form e^x(f(x)+f `(x)). Integrate and get the answer.