If f(a + b - x) = f(x), then _a^b x ,f(x) ,dx = - Vidyadip

MCQ

If $f(a + b - x) = f(x),$ then $\int_a^b {x\,f(x)\,dx = } $

A
$\frac{{a + b}}{2}\int_a^b {f(b - x)\,dx} $
✓
$\frac{{a + b}}{2}\int_a^b {f(x)\,dx} $
C
$\frac{{b - a}}{2}\int_a^b {f(x)\,dx} $
D
None of these

✓

Answer

Correct option: B.

$\frac{{a + b}}{2}\int_a^b {f(x)\,dx} $

b
(b) Since $I = \int_a^b {xf(x)dx = \int_a^b {(a + b - x)f(a + b - x)dx} } $

==> $I = \int_a^b {(a + b)} f(x)dx - \int_a^b {xf(x)dx} $

$\left\{ \because f(a+b-x)=f(x)\text{ given} \right\}$

==> $2I = (a + b)\int_a^b {f(x)dx} $

==> $I = \int_a^b {x\,f(x)dx = \frac{{a + b}}{2}\int_a^b {f(x)dx} } $.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let $a_{1}, a_{2} \ldots, a_{n}$ be a given $A.P.$ whose common difference is an integer and $S _{ n }= a _{1}+ a _{2}+\ldots+ a _{ n }$ If $a_{1}=1, a_{n}=300$ and $15 \leq n \leq 50,$ then the ordered pair $\left( S _{ n -4}, a _{ n -4}\right)$ is equal to

If $\sum_{r=1}^{n}r^3 -\sum_{p=1}^{n}\sum_{m=1}^{p}\sum_{r=1}^{m}1=80$, then possible value of $n$ can be-

From the point $(-1, -60)$ two tangents are drawn to the parabola ${y^2} = 4x$. Then the angle between the two tangents is .................. $^o$

If the angle between the lines whose direction ratios are $2,-1 , 2$ and $a, 3, 5$ be $45^\circ $, then $a =$

If $g(f(x)) = |\sin x|$ and $f(g(x)) = (\sin \sqrt x )^2$, then

Suppose A and B are the coefficients of $30^{\text {th }}$ and $12^{\text {th }}$ terms respectively in the binomial expansion of $(1+x)^{2 n-1}$. If $2 A=5 B$, then $n$ is equal to:

$\alpha$, $\beta$ ,$\gamma$ are roots of equatiuon $x^3 -x -1 = 0$ then equation whose roots are $\frac{1}{{\beta + \gamma }},\frac{1}{{\gamma + \alpha }},\frac{1}{{\alpha + \beta }}$ is

Let the image of the point $(1,0,7)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ be the point $(\alpha, \beta, \gamma)$. Then which one of the following points lies on the line passing through $(\alpha, \beta, \gamma)$ and making angles $\frac{2 \pi}{3}$ and $\frac{3 \pi}{4}$ with $y-$ axis and $z-$ axis respectively and an acute angle with $x -$ axis?

If position vectors of a point $A$ is $a + 2b$ and a divides $ AB $ in the ratio $2:3$, then the position vector of $ B$ is

The area bounded by the curves ${y^2} - x = 0$ and $y - {x^2} = 0$ is