Why doesn't a constant appear when solving $\int{e^x \sin(x)dx}$?

Question

$\int e^x\sin(x)dx$

$= e^x\sin(x) - \int e^x\cos(x)dx$ $\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$(integration by parts with $u = \sin(x) , v' = e^x$ )

$= e^x\sin(x) - (e^x\cos(x)-\int-e^x\sin(x)dx)$ $\space\space\space\space\space\space\space\space$(integration by parts with $u = \cos(x) , v' = e^x$ )

$= e^x(\sin(x)-\cos(x))-\int e^x\sin(x)dx$

$\implies \int e^x\sin(x)dx = e^x(\sin(x)-\cos(x))-\int e^x\sin(x)dx$

$\iff 2\int e^x\sin(x)dx = e^x(\sin(x)-\cos(x))$

$\iff \int e^x\sin(x)dx = \frac{e^x(\sin(x)-\cos(x))}{2}$

But this can't be correct. Why is the $+C$ missing?

Answer 1

Because you just didn't put it. Right in the first step it should be

$$\int e^x \sin(x)\,dx=e^x\sin(x) +C - \int e^x \cos(x) \,dx =...$$

And every time you use partial integration again, there should be another constant to be added.