_ ^ ec x x 2 ;dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{{\cos {\rm{ec}}x}}{{\log \tan \frac{x}{2}}}\;dx = } $

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

✓

Answer

Correct option: A.

$\log \left( {\log \tan \frac{x}{2}} \right) + c$

a
(a) $\log \tan \frac{x}{2} = t $ $ \Rightarrow \frac{1}{{\tan \frac{x}{2}}}.\frac{1}{2}{\sec ^2}\frac{x}{2}\,dx = dt$
$ \Rightarrow {\rm{cosec}}\,x\,dx = dt,$
therefore $\int_{}^{} {\frac{{{\rm{cosec}}\,x}}{{\log \tan \frac{x}{2}}}\,dx} = \int_{}^{} {\frac{1}{t}dt} = \log t + c = \log \left( {\log \tan \frac{x}{2}} \right) + c$.

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

The equation $\sqrt {(x + 1)} - \sqrt {(x - 1)} = \sqrt {(4x - 1)} $ has

If $A( - a,0)$ and $B(a,0)$ are two fixed points, then the locus of the point on which the line $AB$ subtends the right angle, is

Let $x , y$ and $z$ be positive real numbers. Suppose $x , y$ and $z$ are lengths of the sides of a triangle opposite to its angles $X , Y$ and $Z$, respectively. If

$\tan \frac{X}{2}+\tan \frac{Z}{2}=\frac{2 y}{x+y+z},$

then which of the following statements is/are $TRUE$?

$(A)$ $2 Y = X + Z$ $(B)$ $Y=X+Z$ $(C)$ $\tan \frac{x}{2}=\frac{x}{y+z}$ $(D)$ $x^2+z^2-y^2=x z$

If $\vec a$ and $\vec b$ are non-zero vectors which are linearly dependent such that $\frac{{\left| {\vec a + \vec b} \right|}}{{\left| {\vec a - \vec b} \right|}}\, = \,2,\,\left| {\vec b} \right|\, > \,\left| {\vec a} \right|$ Then

If ${\log _{0.04}}(x - 1) \ge {\log _{0.2}}(x - 1)$ then $x$ belongs to the interval

If $\int_{}^{} {\frac{1}{{(\sin x + 4)(\sin x - 1)}}\;dx = A\frac{1}{{\tan \frac{x}{2} - 1}} + B{{\tan }^{ - 1}}(f(x))} + C$, then

If $a, b$ are positive real numbers such that the lines $a x+9 y=5$ and $4 x+b y=3$ are parallel, then the least possible value of $a +b$ is

If $a,\;b,\;c$ are in $G.P.$ and $\log a - \log 2b,\;\log 2b - \log 3c$ and $\log 3c - \log a$ are in $A.P.$, then $a,\;b,\;c$ are the length of the sides of a triangle which is

Let $z$ be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum value of $|\mathrm{z}-(6+8 \mathrm{i})|$ is equal to :

The third term of a $G.P.$ is the square of first term. If the second term is $8$, then the ${6^{th}}$ term is