problem with integeration by parts

Question

I face a problem with this integral. $$\int x\, e^{2x} \, dx = \frac{xe^{2x}}{2} - \frac{1}{2} \int e^{2x} \, dx = \frac{xe^{2x}}{2} - \frac{1}{2}(e^{2x} \, x - 2 \int x \,e^{2x} \, dx)$$ At end the two integral at each side will be omitted(I don't what is the word for this, my English is not very good.) $$\frac{x.e^{2x}}{2} - \frac1{2} \, e^{2x} \, x + \int x \, e^{2x} \, dx = \int x \, e^{2x} \, dx$$ the two sides are the same I don't why this happened.

Answer 1

I give an answer, since I think we misunderstand each other in the comments. The calculation can go like $$ \int xe^{2x}\,dx=x\frac{e^{2x}}{2}-\int 1\frac{e^{2x}}{2}\,dx=x\frac{e^{2x}}{2}-\frac{e^{2x}}{4}+C. $$ In the first step I integrated by parts. In the second step, I only used the fact that $\int e^{2x}\,dx=e^{2x}/2+C$.

What you did in your calculation was that you first integrated by parts, and then reversed it by integrating by parts again (with the roles interchanged). This is why it did not lead you anywhere.