Answer
$$\int ye^{0.2y}dy=5ye^{0.2y}-25e^{0.2y}+C$$
Work Step by Step
$$A=\int ye^{0.2y}dy$$
We would choose $u=y$ and $dv=e^{0.2y}dy$
We could easily see that $du=dy$.
For $dv=e^{0.2y}dy$, we analyze as follows $$\int e^{0.2y}dy=\int\frac{1}{0.2}e^{0.2y}d(0.2y)=5\int e^{0.2y}d(0.2y)=5e^{0.2y}+C$$
Therefore, for $dv=e^{0.2y}dy$, $v=5e^{0.2y}$
Apply Integration by Parts to A, we have $$A=uv-\int vdu$$ $$A=5ye^{0.2y}-\int5e^{0.2y}dy$$ $$A=5ye^{0.2y}-5\int e^{0.2y}dy$$ $$A=5ye^{0.2y}-5(5e^{0.2y}+C)$$ (as we have analyzed above) $$A=5ye^{0.2y}-25e^{0.2y}+C$$
