Evaluate the following integrals: x^2+1 x^4+x^2+1 dx - Vidyadip

Question

Evaluate the following integrals:
$\int\frac{\text{x}^2+1}{\text{x}^4+\text{x}^2+1}\ \text{dx}$

✓

Answer

$\text{I}=\int\frac{\text{x}^2+1}{\text{x}^4+\text{x}^2+1}\ \text{dx}$
$=\int\frac{1+\frac{1}{\text{x}^2}}{\text{x}^2+1+\frac{1}{\text{x}^2}}\ \text{dx}$
Dividing numerator and denominator by $x^2$
$=\int\frac{\Big(1+\frac{1}{\text{x}^2}\Big)}{\Big(\text{x}-\frac{1}{\text{x}}\Big)+3}\ \text{dx}$
Let $\text{x}-\frac{1}{\text{x}}=\text{t}\Rightarrow\Big(1+\frac{1}{\text{x}^2}\Big)\text{dx}=\text{dt}$
$\therefore\text{I}=\frac{\text{dt}}{\text{t}^2+3}$
$=\frac{1}{\sqrt{3}}\tan^{-1}\Big(\frac{\text{t}}{\sqrt{3}}\Big)+\text{C}$
$\therefore\text{I}=\frac{1}{\sqrt{3}}\tan^{-1}\Big(\frac{\text{x}^2-1}{\sqrt{3}\text{x}}\Big)+\text{C}$

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 "Solve the Following Question.(5 Marks)" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Evaluvate the following intregals:
$\int\frac{1}{1-\cot\text{x}}\text{ dx}$

Evaluate the following integrals:
$\int\cos(\log\text{x})\text{dx}$

Without using the concept of inverse of a matrix, find the matrix $\begin{bmatrix}\text{x}&\text{y}\\\text{z}&\text{u}\end{bmatrix}$ such that $\begin{bmatrix}5&-7\\-2&3\end{bmatrix}\begin{bmatrix}\text{x}&\text{y}\\\text{z}&\text{u}\end{bmatrix}=\begin{bmatrix}-16&-6\\7&2\end{bmatrix}$

Find the inverse of the following matrices by using elementry row transformation:$\begin{bmatrix}5 & 2 \\ 2 & 1 \end{bmatrix}$

A manufacturer has three machine I, II, III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for atleast 5 hours a day. She produces only two items M and N each requiring the use of all the three machines.
The number of hours required for producing 1 unit each of M and N on the three machines are given in the following table:

Item	Number of hours required on machines
	I	II	III
M	1	2	1
N	2	1	1.25

She makes a profit of Rs. 600 and Rs. 400 on items M and N respectively. How many of each item should she produce so as to maximise her profit assuming that she can sell all the items that she produced? What will be the maximum profit?

A letter is known to have come either from LONDON or CLIFTON. On the envelope just two consecutive letters ON are visible.
What is the probability that the letter has come from,CLIFTON?

using interation, find the area of the region bounded by the triangle ABC vertices A, B, C are (-1, 1), (0, 5) and (3, 2) respectivrly.

Find the inverse of the following matrices:$\begin{bmatrix}0 & 0 & -1 \\ 3 & 4 & 5 \\ -2 & -4 & -7 \end{bmatrix}$

If $\text{y}=\cos^{-1}(2\text{x})+2\cos^{-1}\sqrt{1-4\text{x}^2}, -\frac{1}{2}<\text{x}<0,$ find $\frac{\text{dy}}{\text{dx}}.$

Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1&\text{a}&\text{a}^2-\text{bc}\\1&\text{b}&\text{b}^2-\text{ac}\\1&\text{c}&\text{c}^2-\text{ab} \end{vmatrix}$