Evaluate the following integrals: ^ 1 2 _ 0 1 (1+x^2) 1-x^2 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\limits^{\frac{1}{2}}_{0}\frac{1}{(1+\text{x}^2)\sqrt{1-\text{x}^2}}\text{ dx}$

✓

Answer

$\text{I}=\int^\limits{\frac{1}{2}}_{0}\frac{1}{(1+\text{x}^2)\sqrt{1-\text{x}^2}}\text{ dx}$
Let $\text{x}=\sin\text{u}$
$\text{dx}=\cos\text{u du}$
$\text{I}=\int^\limits{\frac{\pi}{6}}_{0}\frac{1}{(1+\sin^2\text{u})}\text{ du}$
$\text{I}=\int^\limits{\frac{\pi}{6}}_{0}\frac{\sec^2\text{u}}{(1+2\tan^2\text{u})}\text{ du}$
Let $\tan\text{u}=\text{v}$
$\text{dv}=\sec^2\text{u du}$
$\text{I}=\int^\limits{\frac{1}{\sqrt{3}}}_{0}\frac{1}{(1+2\text{v}^2)}\text{ dv}$
$\text{I}=\frac{1}{\sqrt{2}}\Big[\tan^{-1}\big(\sqrt{2}\text{v}\big)\Big]^{\frac{1}{\sqrt{3}}}_0$
$\text{I}=\frac{1}{\sqrt{2}}\Big[\tan^{-1}\Big(\sqrt{\frac{2}{3}}\Big)\Big]$

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 Definite Integrals→Definite Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Write the vector equation of the following lines and hence determine the distance between them $\frac{\text{x}-1}{2}=\frac{\text{y}-2}{3}=\frac{\text{z}+4}{6}$ and $\frac{\text{x}-3}{4}=\frac{\text{y}-3}{6}=\frac{\text{z}+5}{12}$

Evaluate the following intregals:
$\int\frac{1}{(\sin\text{x}-2\cos\text{x})(2\sin\text{x}-\cos\text{x})}\ \text{dx}$

From the point P (1, 2, 4), a perpendicular is drawn on the plane 2x + y – 2z + 3 = 0. Find the equation, the length and the coordinates of the foot of the perpendicular.

A publisher sells a hard cover edition of a text book for $Rs. 72.00$ and paperback edition of the same ext for $Rs. 40.00.$ Costs to the publisher are $Rs. 56.00$ and $Rs. 28.00$ per book respectively in addition to weekly costs of $Rs. 9600.00.$ Both types require $5$ minutes of printing time, although hardcover requires $10$ minutes binding time and the paperback requires only $2$ minutes. Both the printing and binding operations have $4,800$ minutes available each week. How many of each type of book should be produced in order to maximize profit?

Find $\frac{\text{dy}}{\text{ dx}} $in the following:
$\text{y}=\sin^{-1}\Bigg(\frac{2\text{x}}{1+\text{x}^{2}}\Bigg)$

Form the differential equation of all the circle which pass through the origin and whose centres lies in $y-$axis.

Coloured balls are distributed in three bags as shown in the following table:

Bag	Colour of the ball
Bag	Black	White	Red
$I$	$1$	$2$	$3$
$II$	$2$	$4$	$1$
$III$	$4$	$5$	$3$

A bag is selected at random and then two balls are randomly drawn from the selected bag. They happen to be black and red. What is the probability that they came from bag $I?$

Evaluate the following integrals:
$\int\text{x}\sqrt{\text{x}^4+1}\text{dx}$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1&43&6\\7&35&4\\3&17&2\end{vmatrix}$

Solve the following differential equation:$+ y = \cos x - \sin x.$