Let f (x) = x x , then _0^ 2 f(x) , ,f ( 2 - x ) ,dx = - Vidyadip

MCQ

Let $f (x) = \frac{{\sin x}}{x}$ , then $\int\limits_0^{\frac{\pi }{2}} {f(x)\,\,f\left( {\frac{\pi }{2} - x} \right)\,dx} =$

✓
$\frac{2}{\pi }\int\limits_0^\pi {f(x)\,dx} $
B
$\int\limits_0^\pi {f(x)\,dx} $
C
$\pi \,\int\limits_0^\pi {f(x)\,dx} $
D
$\frac{1}{\pi }\,\int\limits_0^\pi {f(x)\,dx} $

✓

Answer

Correct option: A.

$\frac{2}{\pi }\int\limits_0^\pi {f(x)\,dx} $

a
$I = \int\limits_0^{\frac{\pi }{2}} {\frac{{\sin x\cos x}}{{x\left( {\frac{\pi }{2} - x} \right)}}\,dx} $ $= \int\limits_0^{\frac{\pi }{2}} {\frac{{\sin 2x}}{{x(\pi - 2x)}}\,dx} $; put $2x = t$
$I = \int\limits_0^\pi {\frac{{\sin t}}{{t(\pi - t)}}dt} $ $= \frac{1}{\pi }\,\int\limits_0^\pi {\left( {\frac{{\sin t}}{t} + \frac{{\sin t}}{{(\pi - t)}}} \right)\,dt} $ $=\frac{2}{\pi }\,\int\limits_0^\pi {\frac{{\sin t}}{t}\,dt} $

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

More "M.C.Q (1 Marks)" from STD 12 - 7.2 definite integral→STD 12 - 7.2 definite integral — all question types→Mathematics STD 12 Science question papers→CBSE Board STD 12 Science question papers→CBSE Board question paper generator→

Similar questions

$\int\frac{\text{x}^9}{(4\text{x}^2+1)^6}\text{dx}$ is equal to:

The number of reflexive relations of a set with four elements is equal to

Let $X_1$ and $X_2$ are optimal solutions of a $\text{LPP},$ then:

The area bounded by the curve $y = 4x - x^2$ and the $x-$ axis is :

$\int_{}^{} {\frac{{dx}}{{\tan x + \cot x}}} = $

$A $ $10\,cm$ long rod $ AB$ moves with its ends on two mutually perpendicular straight lines $OX$ and $OY$ . If the end $ A$ be moving at the rate of $2\,cm/\sec $, then when the distance of $A$ from $ O $ is $8\,cm$, the rate at which the end $B$ is moving, is

If the polynomial equation $\text{a}_0\text{x}^{\text{n}}+\text{a}_{\text{n}-1}\text{x}^{\text{n}-1}+\text{a}_{\text{n}-2}\text{x}^{\text{n}-2}+...\text{a}_2\text{x}^2+\text{a}_1\text{x}+\text{a}_0=0$ n positive integer,has two different real roots $\alpha$ and $\beta,$ then between $\alpha$ and $\beta,$ the equation $\text{n}\text{a}_{\text{n}}\text{x}^{\text{n}-1}+(\text{n}-1)\text{a}_{\text{n}-1}\text{x}^{\text{n}-2}+...+\text{a}_1=0$ has:

Let $\vec a = 3\hat i + 2\hat j + 2\hat k$ and $\vec b = \hat i + 2\hat j - 2\hat k$ be two vectors. If a vector perpendicular to both the vectors $\vec a + \vec b$ and $\vec a - \vec b$ has the magnitude $12$ then one such vector is

If $z = {y \over x}\left[ {\sin {x \over y} + \cos \left( {1 + {y \over x}} \right)} \right]$, then $x{{\partial z} \over {\partial x}} = $

Which of these is equal to $\int_0^1\left\{e^x+\sin \frac{\pi x}{4}\right\} d x$ ?