Question

If $\int u\frac{d^{2}v}{d{x}^2}dx=u\frac{dv}{dx}-v\frac{du}{dx}+w$ then $w$ is equal to

• A. $\int v\frac{d^{2}u}{d{x}^2}dx$
• B. $\int u\frac{d^{2}v}{dx}dx$
• C. $\int v{\left(\frac{du}{dx}\right)}^{2}dx$
• D. $\int u{\left(\frac{dv}{dx}\right)}^{2}dx$

Answer 1

Answer: (A)

Differentiate the given integral with respect to $x$
$u\frac{d^{2}v}{d{x}^2}=u\frac{d^{2}v}{d{x}^2}+\frac{du}{dx}\cdot \frac{dv}{dx}-v\frac{d^{2}u}{d{x}^2}-\frac{dv}{dx}\cdot \frac{du}{dx}+\frac{dw}{dx}\Rightarrow \frac{dw}{dx}=v\frac{d^{2}u}{d{x}^2}$
$\therefore dw=v\frac{d^{2}u}{d{x}^2}dx$
$\therefore w=\int v\frac{d^{2}u}{d^2}dx$