Let

1. Use integration by parts to show that

   $$2I=e^x(\sin (x)+\cos (x))$$

2. Hence, find the equation of the curve which passes through the origin and satisfies

   $$\frac{dy}{dx}=e^x\cos (x)$$

   giving your answer in the form

   $$y=R{e}^x\sin (x+\alpha )+c$$

   where $R>0$ and $0<\alpha <\frac{pi}{4}.$