Evan's Trick

A technique to save time for Integration by Parts by adding a constant to $v$

Example

$\int x\arctan (x)dx$ Choose $u=\arctan (x),v^{\prime }=x$ $⟹u^{\prime }=\frac{1}{x^2+1}dx$ $⟹v=\frac{1}{2}{x}^2+\frac{1}{2}$ (Important step, we added the constant $\frac{1}{2}$) Apply Integration by Parts $=(\frac{1}{2}{x}^2+\frac{1}{2})\arctan (x)-\int (\frac{1}{2}{x}^2+\frac{1}{2})(\frac{1}{x^2+1})dx$ $=\frac{1}{2}\arctan (x)x^2+\frac{1}{2}\arctan (x)-\int \frac{1}{2}(x^2+1)(\frac{1}{x^2+1})dx$ $=\frac{1}{2}\arctan (x)x^2+\frac{1}{2}\arctan (x)-\int \frac{1}{2}(1)dx$ $=\frac{1}{2}\arctan (x)x^2+\frac{1}{2}\arctan (x)-\int \frac{1}{2}(1)dx$ $=\frac{1}{2}\arctan (x)x^2+\frac{1}{2}\arctan (x)-\frac{1}{2}x$