Integration by parts
Example 1:
$$\int xe^x \, dx $$
Solution:
$$\int xe^x \, dx $$
$$\textup{Let}\,\, u=x \;\;\;\;\;\;\;\; dv = e^x \,dx$$
$$\;\; du=dx \;\;\;\;\;\;\;\; v = e^x $$
$$\because\int u\,dv = uv-\int v\,du$$
$$\therefore\int xe^x \,dx = xe^x-\int e^x\,dx$$
$$\because\int e^x\,dx = e^x + C$$
$$\therefore\int xe^x \,dx = xe^x-e^x+C$$