If $\displaystyle \int { \left( u\cfrac { dv }{ dx } \right) } dx=uv-\int { wdx }$, then $w=$

Given,

$\displaystyle \int \left ( u\dfrac{dv}{dx} \right )dx=uv-\int wdx$

it is called as, product rule integration,

$\displaystyle \int u.v \ dx=u\int v\ dx$ $\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx$

$\displaystyle \int \left ( u\dfrac{dv}{dx} \right )dx$

$\displaystyle =uv-\int v\dfrac{du}{dx}dx$

$\displaystyle \therefore w=v\dfrac{du}{dx}$