Question

Solve the differential equation \[ \frac{dy}{dx} = e^x \sin x. \]

Hint:

For \( \int e^x f(x) dx \), use integration by parts twice to find the solution.

Answer 1

Step 1: Integrate both sides. \[ y = \int e^x \sin x \,dx. \] Step 2: Use integration by parts: Let \( I = \int e^x \sin x \,dx \). Using ILATE rule, let \( u = \sin x \), \( dv = e^x dx \). Applying integration by parts twice, \[ I = \frac{e^x (\sin x - \cos x)}{2} + C. \]