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 7.2 definite integral→7.2 definite integral — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Function $\text{f}(\text{x})=\cos\text{x}-2\lambda\text{x}$ is monotonic decreasing when:

$\lambda>\frac{1}{2}$
$\lambda<\frac{1}{2}$
$\lambda<2$
$\lambda>2$

The solution of the differential equation $\frac{\text{dy}}{\text{dx}}=1+\text{x}+\text{y}^{2}+\text{xy}^{2}, \text{y}=(0)$ is:

$\text{y}^{2}=\text{exp}\big(\text{x}+\frac{\text{x}^{2}}{2}-1\big)$
$\text{y}^{2}=1+\text{C}\ \text{exp}\Big(\text{x}+\frac{\text{x}^{2}}{2}\Big)$
$\text{y}=\tan (\text{C}+\text{x}+\text{x}^{2})$
$\text{y}=\tan\Big(\text{x}+\frac{\text{x}^{2}}{2}\Big)$

The angle between the lines whose direction cosines are connected by the relations $l + m + n = 0$ and $2lm + 2nl - mn = 0$, is

Let $f: R \rightarrow R$ be given by $f(x)=(x-1)(x-2)(x-5)$. Define $F(x)=\int_0^x f(t) d t, x>0$. Then which of the following options is/are correct?

$(1)$ F has a local minimum at $x=1$

$(2)$ $F$ has a local maximum at $x=2$

$(3)$ $F ( x ) \neq 0$ for all $x \in(0,5)$

$(4)$ F has two local maxima and one local minimum in $(0, \infty)$

A ladder is resting with the wall at an angle of ${30^o}$. A man is ascending the ladder at the rate of $ 3 \,ft/sec$. His rate of approaching the wall is

Choose the correct answer out of the given four options.

Let T be the set of all triangles in the Euclidean plane and let a relation R on T be defined as aRb, if a is congruent to $\text{b}\ \forall\ \text{a},\ \text{b}\in\text{T}.$ Then, R is:

Reflexive but not transitive.
Transitive but not symmetric.
Equivalence.
None of these.

The limit $\lim _{x \rightarrow \infty} x^2 \int \limits_0^x e^{t^3-x^3} d t$ equals

The function $f:R \to \left[ { - \frac{1}{2},\frac{1}{2}} \right],$ defined as $f\left( x \right) = \frac{x}{{1 + {x^2}}}$ is

The distance, of the point $(7,-2,11)$ from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :

Choose the correct option from given four options:
$\int\frac{\text{x}}{\text{x}+1}$ is equal to:

$\text{x}+\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
$\text{x}+\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1-\text{x}|+\text{C}$
$\text{x}-\frac{\text{x}^2}{2}-\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$
$\text{x}-\frac{\text{x}^2}{2}+\frac{\text{x}^3}{3}-\log|1+\text{x}|+\text{C}$