Question

i want to proof one integral problem intgeral by parts integral e^x(sinx+cosx)dx

Answer 1

$Let\ \frac{dv}{dx}=e^x,\ v=e^x,\ Sinx+Cosx=u\\\\I= \int\limits^a_b {e^x(Sin x+Cosx)} \, dx =(Sinx+Cosx)e^x- \int\limits^a_b {(Cosx-Sinx)e^x} \, dx \\\\ =(Sinx+Cosx)e^x- [ (Cosx-Sinx)e^x- \int\limits^a_b {(-Sinx-Cosx)e^x} \, dx ]\\\\(Sinx+Cosx)e^x- (Cosx-Sinx)e^x- I \\\\2*I=2*Sinx* e^x\\\\I=e^x\ Sin\ x+K$