If $f\,\,\left( x \right) = f\,\,\left( {a + b - x} \right)$ for all $x\,\, \in \,\,\left[ {a,\,\,b} \right]$ and $\mathop {\smallint \,}\limits_a^b x{\rm{ }}\,f\,\left( x \right)\,\,dx = k\,\mathop \smallint \limits_a^b \,\,f\left( x \right)\,\,dx$, then the value of $k$ is

$\frac{{{a^2}\,\, + \,\,{b^2}}}{2}$

$\frac{{{a^2}\,\, - \,\,{b^2}}}{2}$

JEE MATHEMATICS MCQ: DEFINITE INTEGRATION

1 Easy

1 of 1

Application +4 | -1

f\,\,\left( x \right) = f\,\,\left( {a + b - x} \right)

for all

x\,\, \in \,\,\left[ {a,\,\,b} \right]

and

\mathop {\int \,}\limits_a^b x\mathrm{ }\,f\,\left( x \right)\,\,dx = k\,\int \limits_a^b \,\,f\left( x \right)\,\,dx

, then the value of

k

View answer and explanation

Correct option: A

\frac{{a\,\, + \,\,b}}{2}\,

Option A is correct because by using the given property and substituting, the value of k is found to be the average of a and b.