Definite Integrals!

$\text{As per the question }$ $\displaystyle \int_0^2 f(x)dx=\cfrac { 3 }{ 4 }$

$\text{We can also evaluate the above integral using integration by parts:}$

$\displaystyle \int_0^2 1.f(x)dx=f(x). \displaystyle \int_0^2 1dx- \displaystyle \int_0^2 f'(x).xdx$

$\text{Hence }$ $\displaystyle \int_0^2 xf'(x)dx=f(x). \displaystyle \int_0^2 1dx-\displaystyle \int_0^2 f(x)dx$

$=[xf(x)]_0^2-3/4$

$= [2f(2)-0f(0)]-3/4$

$= [2*0]-3/4$
$\text{{Since f(2)=0}}$

$= -3/4$