Mathematics

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


ANSWER

$$v\cfrac { du }{ dx } $$


SOLUTION
Given,

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

it is called as, product rule integration,
$$\displaystyle \int u.v \ dx=u\int v\ dx$$ $$\displaystyle -\int \left ( \dfrac{du}{dx}\int v\ dx \right )dx$$

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

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

$$\displaystyle \therefore w=v\dfrac{du}{dx}$$
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Single Correct Medium Published on 17th 09, 2020
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