Question

How do you integrate xe^(2x)dx? ...?

Answer 1

$\int\limits f(x)g(x)dx= F(x)g(x)-\int\limits F(x)g'(x)dx$

F(x)= the antiderivative of f(x) and g'(x)= the derivative of g(x)

$\int\limits xe^{2x}dx \rightarrow f(x)=e^{2x}, g(x)=x, F(x)=\frac{e^{2x}}{2}, g'(x)=1 \\ \\ \rightarrow \frac{1}{2}xe^{2x}-\frac{1}{2}\int\limits e^{2x}dx \rightarrow \frac{1}{2}xe^{2x}-\frac{1}{4}e^{2x}+C$