e-x-sin-2x-dx-

Question Number 497 by 13/NaSaNa(N)056565 last updated on 25/Jan/15

\int e^{x} \sin 2 x d x

Answered by prakash jain last updated on 16/Jan/15

Integrate by parts

I = \int e^{x} \sin 2 x d x

= e^{x} \frac{- \cos 2 x}{2} + \int e^{x} \frac{\cos 2 x}{2}

= - e^{x} \frac{\cos 2 x}{2} + e^{x} \frac{\sin 2 x}{4} - \int e^{x} \frac{\sin 2 x}{4} d x

= - e^{x} \frac{\cos 2 x}{2} + e^{x} \frac{\sin 2 x}{4} - \frac{1}{4} I + C

\frac{5}{4} I = \frac{e^{x}}{4} {\sin 2 x - 2 \cos 2 x}

I = \frac{e^{x}}{5} (\sin 2 x - 2 \cos 2 x) + C