Download App

Indefinite Integration — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integration1 Mark
MCQ
$\int \frac{x-\sin x}{1-\cos x} \cdot d x=$
  • A
    $x \cot \left(\frac{x}{2}\right)+ C$
  • ✓
    $- x \cot \left(\frac{x}{2}\right)+ c$
  • C
    $\cot \left(\frac{x}{2}\right)+c$
  • D
    $x \tan \left(\frac{x}{2}\right)+ C$
✓

Answer

Correct option: B.
$- x \cot \left(\frac{x}{2}\right)+ c$
$- x \cot \left(\frac{x}{2}\right)+ c$
Hint :
$\int \frac{x-\sin x}{1-\cos x} d x=\int \frac{x-2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}{2 \sin ^2\left(\frac{x}{2}\right)} d x$
$= \frac{1}{2} \int x \operatorname{cosec}^2\left(\frac{x}{2}\right) d x-\int \cot \left(\frac{x}{2}\right) d x$
$ =\frac{1}{2}\left[x \int \operatorname{cosec}^2\left(\frac{x}{2}\right) d x-\int\left[\frac{d}{d x}(x) \int \operatorname{cosec}^2\left(\frac{x}{2}\right) d x\right] d x\right. \\ -\int \cot \left(\frac{x}{2}\right) d x$
$=\frac{1}{2}\left[x\left\{\frac{-\cot \left(\frac{x}{2}\right)}{\left(\frac{1}{2}\right)}\right\}-\int 1 \cdot \frac{-\cot \left(\frac{x}{2}\right)}{\left(\frac{1}{2}\right)} d x-\int \cot \left(\frac{x}{2}\right) d x\right.]$
$=-x \cot \left(\frac{x}{2}\right)+\int \cot \left(\frac{x}{2}\right) d x-\int \cot \left(\frac{x}{2}\right) d x$
$=-x \cot \left(\frac{x}{2}\right)+c.$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Indefinite Integration→Indefinite Integration — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

If the p.d.f. of a c.r.v. $X$ is $f(x)=3\left(1-2 x^2\right)$, for $0<x<1$ and $=0$, otherwise (elsewhere), then the c.d.f. of $X$ is $F(x)=$
→
If $\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{\pi}{2}$, then the value of $x^2+y^2+z^2+2 x y z$ is equal to
→
The solution of the differential equation $xdy + ydy = x^2y dy - y^2x dx$, is:
→
If the set $A$ contains $7$ elements and the set $B$ contains $10$ elements, then the number one$-$one functions from $A$ to $B$ is:
→
The angle between the lines given by the equation $\lambda y^2+\left(1-\lambda^2\right) x y-\lambda x^2=0$ is
→
If $\frac{d y}{d x}=\tan ^2(x+y)$, then $\sin 2(x+y)=$
→
The direction ratios of the line perprndicular to the lines $\frac{\text{x}-7}{2}=\frac{\text{y}+17}{-3}=\frac{\text{z}-6}{1}$ and, $\frac{\text{x}+5}{1}=\frac{\text{y}+3}{2}=\frac{\text{z}-4}{-2}$ are proportional to:
→
If $\mathrm{A}=\left[\begin{array}{ll}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha\end{array}\right]$ then $\mathrm{A}^{-1}=\ldots$
→
Equation of the plane passing through $(-1,3,4)$ and parallel to $Y Z$-plane is
→
The eqution of the plane $\vec{\text{r}}=\hat{\text{i}}-\hat{\text{j}}+\lambda(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+\mu(\hat{\text{i}}-2\hat{\text{j}}+3\hat{\text{k}})$ in scalar product from is:
→