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 "4 Marks" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Find: $\int\frac{3\text{x}+5}{\text{x}^2+3\text{x}-18}\text{ dx}.$

Evaluate the following definite integrals:
$\int_{\text{e}}^\limits{\text{e}^2}\Big\{\frac{1}{\log\text{x}}-\frac{1}{(\log\text{x})^2}\Big\}\text{dx}$

$\text{Find:}\int \frac{\text{x}\ \text{dx}}{(2 + \text{x}^{\text{2}}) (4 + \text{e}^{4\text{}})}$

Find the inverse of the following matrices:

$\begin{bmatrix}0 & 1 & -1 \\ 4 & -3 & 4 \\ 3 & -3 & 4 \end{bmatrix}$

Using matrices, solve the following system of equations:

4x + 3y + 2z = 60.

x + 2y + 3z = 45.

6x + 2y + 3z = 70.

Express the matrix $\begin{bmatrix}2&3&1\\1&-1&2\\4&1&2\end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

The prices of three commodities P, Q and R are Rs. x, y and z per unit respectively. A purchases 4 units of R and sells 3 units of P and 5 units of Q. B purchases 3 units of Q and sells 2 units of P and 1 unit of R. C purchases 1 unit of P and sells 4 units of Q and 6 units of R. In the process A, B and C earn Rs. 6000, Rs. 5000 and Rs. 13000 respectively. If selling the units is positive earning and buying the units is negative earnings, find the price per unit of three commodities by using matrix method.

Prove the following identities:
$\begin{vmatrix}\text{y}+\text{z}&\text{z}&\text{y}\\\text{z}&\text{z}+\text{x}&\text{x}\\\text{y}&\text{x}&\text{x}+\text{y}\end{vmatrix}=4\text{xyz}$

One kind of cake requires 300 g of flour and 15 g of fat, another kind of cake requires 150 g of flour and 30 g of fat. Find the maximum number of cakes which can be made from 7.5 kg of flour and 600 g of fat, assuming that there is no shortage of the other ingradients used in making the cakes. Make it as an L.P.P. and solve it graphically.

Maximum Z = x - 5y + 20
Subject to
$\text{x}-\text{y}\geq0$
$-\text{x}+2\text{y}\geq2$
$\text{x}\geq3$
$\text{y}\geq4$
$\text{x},\text{y}\geq0$