Question

If ${\int }_0^x\varphi \left(x\right)dx=x+{\int }_x^{2}x\varphi \left(x\right)dx$, then $\varphi \left(4\right)$ equals.

• A. $\frac{1}{3}$
• B. $\frac{1}{4}$
• C. $\frac{1}{5}$
• D. $1$

Answer 1

Answer: (C)

$\frac{1}{5}$

${\int }_0^x\varphi \left(x\right)dx=x+{\int }_x^{2}x\varphi \left(x\right)dx$
By Leibnitz Rule, Differentiating both sides with respect to $x$,
$\varphi \left(x\right)=1+(0-x\varphi (x\left)\right)$.
$\Rightarrow \varphi \left(x\right)=\frac{1}{1+x}\to \varphi \left(4\right)=\frac{1}{1+4}=\frac{1}{5}$.