If f (a + b - x) = f (x) , then _{a}^{b} x f (x) dx is equal to

Correct Answer is : a+b2∫aba+b/2_{a}^{b}2a+ba∫b f (x) dx

Correct Answer is : a+b2∫aba+b/2_{a}^{b}2a+ba∫b f (x) dx

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VITEEE 2016 Paper

Section: Aptitude

Question : 120 of 125

Marks: +1, -0

\int\limits_{a}^{b}

x f (x) dx is equal to

Solution:

Let I =

\int\limits_{a}^{b}

x f (x) dx

Let a + b - x = z ⇒ - dx = dz

When x = a , z = b and when x = b , z = a

∴ I = -

\int\limits_{b}^{a}

(a + b - z) f (z) dz

I = (a + b)

\int\limits_{b}^{a}

f (x) dx -

\int\limits_{a}^{b}

x f (x) dx

I = (a + b)

\int\limits_{b}^{a}

f (x) dx - 1 ;

2I = (a + b)

\int\limits_{a}^{b}

f (x) dx

I =

\left(\frac{a+b}{2}\right)\int\limits_{a}^{b}

f (x) dx

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