_0^1 ( 2 x ) ,dx = - Vidyadip

MCQ

$\int_0^1 {\log \sin \left( {\frac{\pi }{2}x} \right)} \,dx = $

✓
$ - \log 2$
B
$\log 2$
C
$\frac{\pi }{2}\log 2$
D
$ - \frac{\pi }{2}\log 2$

✓

Answer

Correct option: A.

$ - \log 2$

a
(a) Put $\frac{\pi }{2}x = \theta \Rightarrow dx = \frac{2}{\pi }d\theta $;

As $x = 0$ to $1,$ $\theta = 0$ to $\frac{\pi }{2}$

Then it reduces to

$\frac{2}{\pi }\int_0^{\pi /2} {\,\,\log \sin \theta \,d\theta = \frac{2}{\pi }\left[ { - \frac{\pi }{2}\log 2} \right]} $

$ = - \log 2$.

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

The function $\text{f(x)=}\begin{cases}\frac{\text{e}\frac{1}{\text{x}}-1}{\text{e}\frac{1}{\text{x}}+1},&\text{x}\neq0\\0,&\text{x}=0\end{cases}$

$\int\frac{1}{\cos\text{x}+\sqrt{3}\sin\text{x}}\text{ dx}$ is equal to:

A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is

Let $*$ be a binary operation on $Q^{+}$ defined by $\text{a}^*\text{b}=\frac{\text{ab}}{100}\forall\text{ a, b}\in\text{Q}^+$. The inverse of $0.1$ is:

If $\vec{\text{a}}$ and $\vec{\text{b}}$ are unit vectors, then which of the following values of $\vec{\text{a}}.\vec{\text{b}}$ is not possible?

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b} \cdot \vec{c}=0$ and $\vec{a} \times(\vec{b} \times \vec{c})=\frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b} \cdot \vec{d}=\vec{a} \cdot \vec{b}$, then $(\vec{a} \times \vec{b}) \cdot(\vec{c} \times \vec{d})$ is equal to

If the function $y = e^{4x} + 2e^{-x}$ is a solution of the differential equation $\frac{{\frac{{{d^3}y}}{{d{x^3}}} - 13\frac{{dy}}{{dx}}}}{y} = K$ then the value of $K$ is

Let $f:\left[-\frac{1}{2}, 2\right] \rightarrow \mathbb{R}$ and $g:\left[-\frac{1}{2}, 2\right] \rightarrow \mathbb{R}$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x| f(x)+|4 x-7| f(x)$, where $[y]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb{R}$. Then

($A$) $f$ is discontinuous exactly at three points in $\left[-\frac{1}{2}, 2\right]$

($B$) $f$ is discontinuous exactly at four points in $\left[-\frac{1}{2}, 2\right]$

($C$) $g$ is $NOT$ differentiable exactly at four points in $\left(-\frac{1}{2}, 2\right)$

($D$) $g$ is $NOT$ differentiable exactly at five points in $\left(-\frac{1}{2}, 2\right)$

Range of $\text{f(x)}=\sqrt{(1-\cos\text{x})\sqrt{(1-\cos\text{x})\sqrt{(1-\cos\text{x}).....\infty}}}$

Let $f : N \rightarrow R : \text{f}(\text{x})=\frac{(2\text{x}-1)}{2}$ and $g : Q \rightarrow R : g(x) = x + 2$ be two functions. Then, $(gof) (\frac{3}{2})$ is: