$\int e^{x}\sin x\,dx$ equals

1. Let $I=\int e^{x}\sin x\,dx$. By parts ($u=e^{x}$): $I=-e^{x}\cos x+\int e^{x}\cos x\,dx$. 2. For $\int e^{x}\cos x\,dx$, by parts: $e^{x}\sin x-\int e^{x}\sin x\,dx=e^{x}\sin x-I$. 3. So $I=-e^{x}\cos x+e^{x}\sin x-I$, giving $2I=e^{x}(\sin x-\cos x)$. 4. Hence $I=\dfrac{e^{x}}{2}(\sin x-\cos x)+C$. Option B has the wrong inner sign. _Source: NCERT Class 12 Mathematics Ch 7 "Integrals", p.327_