_ ,0 ^ , ^2 x ,dx = - Vidyadip

MCQ

$\int_{\,0}^{\,\pi } {\log {{\sin }^2}x\,dx = } $

✓
$2\pi {\log _e}\left( {\frac{1}{2}} \right)$
B
$\pi {\log _e}2 + c$
C
$\frac{\pi }{2}{\log _e}\left( {\frac{1}{2}} \right) + c$
D
None of these

✓

Answer

Correct option: A.

$2\pi {\log _e}\left( {\frac{1}{2}} \right)$

a
(a) $\int_0^\pi {2\log \sin xdx = 2\int_0^{2\frac{\pi }{2}} {\log \sin xdx = 4\int_0^{\pi /2} {\log \sin x\,dx} } } $

$ = 4 \times \left( { - \frac{\pi }{2}\log 2} \right) = - 2\pi {\log _e}2 = 2\pi {\log _e}\left( {\frac{1}{2}} \right)$.

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 integral $\frac{24}{\pi} \int_{0}^{\sqrt{2}} \frac{\left(2-x^{2}\right) d x}{\left(2+x^{2}\right) \sqrt{4+x^{4}}}$ is equal to

If $P(A)=\frac{1}{2}, P(B)=0,$ then $P(A | B)$ is

If $f(x) = A\sin \left( {\frac{{\pi x}}{2}} \right) + B,$ $f'\left( {\frac{1}{2}} \right) = \sqrt 2 $ and $\int_0^1 {f(x)\,dx = \frac{{2A}}{\pi },} $ then the constants $A$ and $B$ are respectively

$\int_{}^{} {\frac{{dx}}{{1 + 3{{\sin }^2}x}} = } $

Let $f (x)$ be a function satisfying $f ' (x) = f (x)$ with $ f (0) = 1$ and $g$ be the function satisfying $f (x) + g (x) = x^2$ . The value of the integral $\int\limits_0^1 {f(x)g(x)\,dx} $ is

For some $a, b, c \in N$, let $f(x)=a x-3$ and $g ( x )= x ^{ b }+ c , x \in R$. If $(\text { fog })^{-1}( x )=\left(\frac{ x -7}{2}\right)^{1 / 3}$ then $(fog) (ac) + (gof) (b)$ is equal to $..........$

If $P(A)=\frac{3}{10}, P(B)=\frac{2}{5}$ and $P(A \cup B)=\frac{3}{5}$, then find the value of $P(B / A)$.

Integrating factor of the differential equation $\frac{d y}{d x}+y \tan x-\sec x=0$ is

The set of points where the function $f(x) = x |x|$ is differentiable is :

The value of $\lambda$ for which $\int {\frac{{4{x^3} + \lambda {4^x}}}{{{4^x} + {x^4}}}} \,\,dx = \log ({4^x} + {x^4}) + c$ is