Download App

Model Paper 10 — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSModel Paper 103 Marks
Question
Prove that: $\int_0^\pi \frac{x}{1-\cos \alpha \sin x} d x=\frac{\pi(\pi-\alpha)}{\sin \alpha}$
✓

Answer

Let $I=\int_0^\pi \frac{x}{1-\cos \alpha \sin x} d x$. Then,
$I=\int_0^\pi \frac{(\pi-x)}{1-\cos \alpha \sin (\pi-x)} d x$
By using property of defnite integrals,
$\Rightarrow I=\int_0^\pi \frac{\pi}{1-\cos \alpha \sin x} d x-\int_0^\pi \frac{x}{1-\cos \alpha \sin x} d x$
$\Rightarrow I=\pi \int_0^\pi \frac{1}{1-\cos \alpha \sin x} d x-I$
$\Rightarrow 2 I=\pi \int_0^\pi \frac{1}{1-\cos \alpha \sin x} d x$
$\Rightarrow 2 I=\pi \int_0^\pi \frac{1+\tan ^2 x / 2}{\left(1+\tan ^2 x / 2\right)-2 \cos \alpha \tan x / 2} d x$
$\Rightarrow 2 I=\pi \int_0^\pi \frac{\sec ^2 x / 2}{\tan ^2 x / 2-2 \cos \alpha \tan x / 2+1} d x$
$\Rightarrow I=\frac{\pi}{2} \int_0^\pi \frac{\sec ^2 x / 2}{\tan ^2 x / 2-2 \cos \alpha \tan x / 2+1} d x$
Let $ \tan \frac{x}{2}=t$. Then, $d\left(\tan \frac{x}{2}\right)=d t \Rightarrow \sec ^2 \frac{x}{2} d x=2 d t$
Also, $ x=0 \Rightarrow t=\tan 0=0$ and $x=\pi \Rightarrow t=\tan \frac{\pi}{2}=\infty$
$\therefore I=\frac{\pi}{2} \int_0^{\infty} \frac{2 d t}{t^2-2 t \cos \alpha+1}$
$\Rightarrow I=\pi \int_0^{\infty} \frac{1}{(t-\cos \alpha)^2+\left(1-\cos ^2 \alpha\right)} d t$
$\Rightarrow I=\pi \int_0^{\infty} \frac{1}{\sin ^2 \alpha+(t-\cos \alpha)^2} d t$
$\Rightarrow I=\frac{\pi}{\sin \alpha}\left[\tan ^{-1}\left(\frac{t-\cos \alpha}{\sin \alpha}\right)\right]_0^{\infty}$
$\Rightarrow I=\frac{\pi}{\sin \alpha}\left[\tan ^{-1} \alpha-\tan ^{-1}(-\cot \alpha)\right]$
$\Rightarrow I=\frac{\pi}{\sin \alpha}\left[\frac{\pi}{2}+\tan ^{-1}(\cot \alpha)\right]$
$\left.\Rightarrow I=\frac{\pi}{\sin \alpha}\left[\frac{\pi}{2}+\tan ^{-1}\left\{\tan \left(\frac{\pi}{2}-\alpha\right)\right)\right\}\right]=\frac{\pi}{\sin \alpha}\left[\frac{\pi}{2}+\frac{\pi}{2}-\alpha\right]=\frac{\pi(\pi-\alpha)}{\sin \alpha}$

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 "3 Marks Question" from Model Paper 10→Model Paper 10 — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Give a condition that three vectors $\vec{\text{a}},\ \vec{\text{b}}\text{ and }\vec{\text{c}}$ from the three sides of a triangle. what are the other possibilities?
→
For what value of k is the function
$\text{f}\text{(x)}=\begin{cases}\frac{\sin5\text{x}}{3\text{x}}, &\text{if}\text{ x}\neq0\\\text{k}, &\text{if}\text{ x}=0\end{cases}$ is continuous at x = 0?
→
If $\vec{\text{a}},\vec{\text{b}}$ are two vectors such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|=\big|\vec{\text{b}}\big|,$ then prove that $\vec{\text{a}}+2\vec{\text{b}}$ is perpendicular to $\vec{\text{a}}.$
→
By using the properties of definite integrals, evaluate the integral $\int\limits_{ - 5}^5 {\left| {x + 2} \right|dx} $
→
Evaluate the following:
$\tan^{-1}1+\cos^{-1}\Big(-\frac{1}{2}\Big)+\sin^{-1}\Big(-\frac{1}{2}\Big)$
→
Five defective bolts are acciedently mixed with twenty good ones. If four bolts are drawn at random from this lot, find the probability distribution of the number of defective bolts.
→
Evaluate the following integrals:
$\int\frac{\sin(2+3\log\text{x})}{\text{x}}\text{ dx}$
→
If $O$ be the origin and the coordinates of $P$ be $(1, 2, -3),$ then find the equation of the plane passing through $P$ and perpendicular to $OP.$
→
Define a symmetric matrix. Prove that for $\text{A}=\begin{bmatrix}2&4 \\5&6 \end{bmatrix}, A + A^T$ is a symmetric matrix where $A^T$ is the transpose of $A$.
→
If $\text{f(x)}=\begin{cases}2\text{x}^2+\text{k},&\text{if }\text{ x}\geq0\\-2\text{x}^2+\text{k},&\text{if }\text{ x}<0\end{cases},$ then what should be the value of k so that f(x) is continuous at x = 0.
→