Can we integrate, the indefinite integral $\int 2aye^{ay^2}dy$ by parts? Can we integrate, the indefinite integral $\int 2aye^{ay^2}dy$ by parts ? If we let, $ay^2= t$, we have $(2ay)dy=dt$ and the integral can be integrated. This process is known as integration by substitution.

Yes, but you're not going to like it.

Let $u=1$, $dv=2aye^{ay^2}\,dy$. Then $du=0dy$, and (by the substitution technique that you already know about) $v=e^{ay^2}$. So $$\int2aye^{ay^2}\,dy=\int u\,dv=uv-\int v\,du=e^{ay^2}-\int0dy=e^{ay^2}$$