MCQ
$\int_{}^{} {\frac{{x - \sin x}}{{1 - \cos x}}dx = } $
- A$x\cot \frac{x}{2} + c$
- ✓$ - x\cot \frac{x}{2} + c$
- C$\cot \frac{x}{2} + c$
- DNone of these
✓
Answer
Correct option: B.
$ - x\cot \frac{x}{2} + c$
b
(b)$\int_{}^{} {\frac{{x - \sin x}}{{1 - \cos x}}\,dx} = \int_{}^{} {\frac{x}{{1 - \cos x}}\,dx} - \int_{}^{} {\frac{{\sin x}}{{1 - \cos x}}\,dx} $
$ = \frac{1}{2}\int_{}^{} {x\,{\rm{cose}}{{\rm{c}}^{\rm{2}}}\left( {\frac{x}{2}} \right)\,dx} - \int_{}^{} {\frac{{2\sin (x2)\cos (x2)}}{{2{{\sin }^2}(x2)}}\,dx} $
$ = \frac{1}{2}\int_{}^{} {x\,{\rm{cose}}{{\rm{c}}^{\rm{2}}}\left( {\frac{x}{2}} \right)\,dx} - \int_{}^{} {\cot \left( {\frac{x}{2}} \right)\,dx} $$ = - x\cot \left( {\frac{x}{2}} \right) + c$.
(b)$\int_{}^{} {\frac{{x - \sin x}}{{1 - \cos x}}\,dx} = \int_{}^{} {\frac{x}{{1 - \cos x}}\,dx} - \int_{}^{} {\frac{{\sin x}}{{1 - \cos x}}\,dx} $
$ = \frac{1}{2}\int_{}^{} {x\,{\rm{cose}}{{\rm{c}}^{\rm{2}}}\left( {\frac{x}{2}} \right)\,dx} - \int_{}^{} {\frac{{2\sin (x2)\cos (x2)}}{{2{{\sin }^2}(x2)}}\,dx} $
$ = \frac{1}{2}\int_{}^{} {x\,{\rm{cose}}{{\rm{c}}^{\rm{2}}}\left( {\frac{x}{2}} \right)\,dx} - \int_{}^{} {\cot \left( {\frac{x}{2}} \right)\,dx} $$ = - x\cot \left( {\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.
Explore more
Similar questions
The solution of the differential equation $x + y\frac{{dy}}{{dx}} = 2y$ is
→In a given figure of square $6\ A's$ should be written in such a manner that every row contains atleast one $'A'.$ It is possible in number of ways is
→The area enclosed by the curves $y = x^2, y = x^3 , x = 0$ and $x = p,$ where $p > 1$ , is $1/6$. The $p$ equals
→If $A = \left[ {\begin{array}{*{20}{c}}5&2\\3&1\end{array}} \right],$then ${A^{ - 1}}$=
→If $\theta $ and $\phi $ are angles in the $1^{st}$ quadrant such that $\tan \theta = 1/7$ and $\sin \phi = 1/\sqrt {10} $.Then
→If three letters can be posted to any one of the $5$ different addresses, then the probability that the three letters are posted to exactly two addresses is:
→The expression $[x + (x^3-1)^{1/2}]^5 + [x - (x^3-1)^{1/2}]^5$ is a polynomial of degree :
→If $\mathrm{a, b, c}$ are in $\mathrm{A.P}$, find value of
→$\left|\begin{array}{ccc}
2 y+4 & 5 y+7 & 8 y+a \\
3 y+5 & 6 y+8 & 9 y+b \\
4 y+6 & 7 y+9 & 10 y+c
\end{array}\right|$
Let $\mathrm{f}: N \rightarrow N$ be a function such that $\mathrm{f}(\mathrm{m}+\mathrm{n})=\mathrm{f}(\mathrm{m})+\mathrm{f}(\mathrm{n})$ for every $\mathrm{m}, \mathrm{n} \in N$. If $\mathrm{f}(6)=18$ then $\mathrm{f}(2) \cdot \mathrm{f}(3)$ is equal to :
→If the points $(0, 0)$, $(2,\,2\sqrt 3 )$ and $(a, b)$ be the vertices of an equilateral triangle, then $(a,\,b) = $
→