Evaluate the following integrals: 1 (1+x^2) 1-x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{1}{(1+\text{x}^2)\sqrt{1-\text{x}^2}}\text{ dx}$

✓

Answer

We have
$\text{I}=\int\frac{1}{(1+\text{x}^2)\sqrt{1-\text{x}^2}}\text{ dx}$
Putting $\text{x}=\frac{1}{\text{t}}$
$\text{dx}=-\frac{1}{\text{t}^2}\text{ dt}$
$\therefore\ \text{I}=\int\frac{-\frac{1}{\text{t}^2}\text{dt}}{\big(1+\frac{1}{\text{t}^2}\big)\sqrt{1-\frac{1}{\text{t}^2}}}$
$=\int\frac{-\frac{1}{\text{t}^2}\text{ dt}}{\frac{(\text{t}^2+1)}{\text{t}^2}\frac{\sqrt{\text{t}^2-1}}{\text{t}}}$
$=-\int\frac{\text{t dt}}{(\text{t}^2+1)\sqrt{\text{t}^2-1}}$
Again Putting $\text{t}^2-1=\text{u}^2$
$2\text{tdt}=2\text{udu}$
$\text{tdt}=\text{udu}$
$\therefore\ \text{I}=-\int\frac{\text{u du}}{(\text{u}^2+2)\text{u}}$
$=-\int\frac{\text{du}}{\text{u}^2+(\sqrt{2})^2}$
$=-\frac{1}{\sqrt{2}}\tan^{-1}\Big(\frac{\text{u}}{\sqrt{2}}\Big)+\text{C}$
$=-\frac{1}{\sqrt{2}}\tan^{-1}\bigg(\frac{\sqrt{\text{t}^2-1}}{\sqrt{2}}\bigg)+\text{C}$
$=-\frac{1}{\sqrt{2}}\tan^{-1}\Bigg(\sqrt{\frac{\frac{1}{\text{x}^2}-1}{2}}\Bigg)+\text{C}$
$=-\frac{1}{\sqrt{2}}\tan^{-1}\bigg(\sqrt{\frac{1-\text{x}^2}{2\text{x}^2}}\bigg)+\text{C}$

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 "5 Marks Questions" from Indefinite Integrals→Indefinite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Determine whether the following pair of lines intersect or not:
$\frac{\text{x}-1}{3}=\frac{\text{y}-1}{-1}=\frac{\text{z}+1}{0}$ and $\frac{\text{x}-4}{2}=\frac{\text{y}-0}{0}=\frac{\text{z}+1}{3}$

Find the points of discontinuity, if any of the following function:
$\text{f(x)}=\begin{cases}|\text{x}-3|,&\text{if }\text{ x}\geq1\\\frac{\text{x}^2}{4}-\frac{3\text{x}}{2}+\frac{13}{4},&\text{if }\text{ x}<1\end{cases}$

Coloured balls are distributed in four boxes as shown in the following table:

Box	Colour
Box	Black	White	Red	Blue
$I$ $II$ $III$ $IV$	$3$ $2$ $1$ $4$	$4$ $2$ $2$ $3$	$5$ $2$ $3$ $1$	$6$ $2$ $1$ $5$

A box is selected at random and then a ball is randomly drawn from the selected box. The colour of the ball is black, what is the probability that ball drawn is from the box $III.$

If $y^x = e^{y–x},$ prove that $\frac{\text{dy}}{\text{dx}} = \frac{(1 + \log\text{y})^{2}}{\log\text{y}}.$

If $ \vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}\ \text{and}\ \vec{\text{b}}=7\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}, $ then express $ \overrightarrow{\text{b}}$ in the form of $\overrightarrow{\text{b}}=\ \overrightarrow{\text{b}}_1+\overrightarrow{\text{b}}_2,$ where $ \overrightarrow{\text{b}}_1$ is parallel to $\overrightarrow{\text{a}}$ and $ \overrightarrow{\text{b}}_2$ is perpendicular to$\overrightarrow{\text{a}}$.

Verify the Rolle’s theorem for each of the functions:
$\text{f(x)}=\log(\text{x}^2+2)-\log3\text{ in }[-1,1].$

If f(2a - x) = -f(x), prove that $\int\limits^{2\text{a}}_0\text{f(x)}\text{dx}=0$

Differentiate the following functions with respect to x:
$\sin^{-1}\Big(\frac{\text{x}}{\sqrt{\text{x}^2+\text{x}^2}}\Big)$

Evaluvate the following intregals:
$\int\frac{3+2\cos\text{x}+4\sin\text{x}}{2\sin\text{x}+\cos\text{x}+3}\ \text{dx}$

An insurance company insured $3000$ scooters, $4000$ cars and $5000$ trucks. The probabilities of the accident involving a scooter, a car and a truck are $0.02, 0.03$ and $0.04$ respectively. One of the insured vehicles meet with an accident. Find the probability that it is a,

Scooter.
Car.
Truck.