Question

If $f\left( a+b-x \right)=f\left( x \right),$ then $\int\limits_{a}^{b}{x\text{ }f\left( x \right)}dx$

(a)$\dfrac{a+b}{2}\int\limits_{a}^{b}{\text{ }f\left( b-x \right)}dx$

(b)$\dfrac{a+b}{2}\int\limits_{a}^{b}{\text{ }f\left( b+x \right)}dx$

(c)$\dfrac{b-a}{2}\int\limits_{a}^{b}{\text{ }f\left( x \right)}dx$

(d)$\dfrac{a+b}{2}\int\limits_{a}^{b}{\text{ }f\left( x \right)}dx$

Answer 1

Hint: Take the internal given as $\text{I}$. Put $\left( \left( a+b \right)-x \right)=x$ in the place of $x$ in the integral. Now open the brackets and simplify it. Add this expression to our integral given. Thus find the value of $\text{I}$ from the integral.

Complete step-by-step answer:
We have been given an integral, for which we need to find the solution.
Let us take integral as $\text{I}$.

$\text{I}=\int\limits_{a}^{b}{x\text{ }f\left( x \right)}dx$ $\to$(1)

We have been given $f\left( a+b-x \right)=f\left( x \right),$ $\to$(2)

One of the basic properties of definite integrals.
$\int\limits_{a}^{b}{x\text{ }f\left( x \right)}dx=\int\limits_{a}^{b}{f\left( a+b-x \right)}dx$ $\to$(3)

By comparing (1) and (2). We can modify equation (1) by putting
$x=a+b-x$In equation (1)

$I=\int\limits_{a}^{b}{\left( a+b-x \right)}dx=\int\limits_{a}^{b}{\left( \left( a+b \right)-x \right)}dx$

Now let us open the bracket and simplify it

$I=\int\limits_{a}^{b}{\left( a+b \right)}f\left( x \right)dx-\int\limits_{a}^{b}{xf\left( x \right)}dx$ $\to$(4)

Now let us add both equation (1) and equation (4)

$I+I=\int\limits_{a}^{b}{xf\left( x \right)}dx+\int\limits_{a}^{b}{\left( a+b \right)}f\left( x \right)dx-\int\limits_{a}^{b}{xf\left( x \right)}dx$

Now let us cancel out the integral $\int\limits_{a}^{b}{xf\left( x \right)}dx$

$2I=\int\limits_{a}^{b}{\left( a+b \right)}f\left( x \right)dx$

Let us simplify this expression further

& 2I=\left( a+b \right)\int\limits_{a}^{b}{f}\left( x \right)dx \\\ & \\\ & I=\left( \dfrac{a+b}{2} \right)\int\limits_{a}^{b}{f}\left( x \right)dx \\\ \end{aligned}$$ Hence we got the integral value as, $$I=\left( \dfrac{a+b}{2} \right)\int\limits_{a}^{b}{f}\left( x \right)dx$$ $$\therefore $$Option (d) is the correct answer. Note: The given expression to us was of definite integral. The property of definite integral that we have used is not commonly asked, but just remember the formula. The integral is direct substitution of the expressions given. Always take the expression as equal to I.