Question

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

• A. $\int v \frac{d^2 u}{d x^2} d x$
• B. $\int u \frac{ d ^2 v }{ dx } dx$
• C. $\int v\left(\frac{d u}{d x}\right)^2 d x$
• 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{d u}{d x} \cdot \frac{d v}{d x}-v \frac{d^2 u}{d x^2}-\frac{d v}{d x} \cdot \frac{d u}{d x}+\frac{d w}{d x} \Rightarrow \frac{d w}{d x}=v \frac{d^2 u}{d x^2} $
$\therefore dw = v \frac{ d ^2 u }{ dx ^2} dx $
$\therefore w=\int v \frac{d^2 u}{d^2} d x $