Evaluate the follwing intregals: 1 x^4-1 dx - Vidyadip

Question

Evaluate the follwing intregals:
$\int\frac{1}{\text{x}^4-1}\text{ dx}$

✓

Answer

The Evaluate the integral follow the steps,
$\int\frac{1}{(\text{x}^4-1)}\text{ dx}$
Let $\frac{1}{(\text{x}^4-1)}=\frac{\text{A}}{\text{x}-1}+\frac{\text{B}}{\text{x}-1}+\frac{\text{C}}{\text{x}^2+1}$
$1={\text{A}}{(\text{x}-1})(\text{x}^2+1)+\text{B}(\text{x}+1)(\text{x}^2+1)+\text{C}(\text{x}+1)(\text{x}-1)$
$\text{For x}=1,\text{ B}=\frac{1}{4}$
$\text{For x}=-1,\text{ A}=-\frac{1}{4}$
$\text{For x}=0,\text{ C}=-\frac{1}{2}$
Therefore,
$\int\frac{1}{(\text{x}^4-1)}\text{ dx}=-\frac{1}{4}\int\frac{\text{dx}}{\text{x}+1}+\frac{1}{4}\frac{\text{dx}}{\text{x}-1}-\frac{1}{2}\int\frac{\text{dx}}{\text{x}^2+1}$
$=-\frac{1}{4}\ln\big|(\text{x}+1)\big|+\frac{1}{4}\ln\big|(\text{x}-1)\big|-\frac{1}{2}\tan^{-1}\text{x}+\text{C}$
$=\frac{1}{4}\ln\Big|\frac{\text{x}-1}{\text{x}+1}\Big|-\frac{1}{2}\tan^{-1}\text{x}+\text{C}$

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

Similar questions

Integrate the function: $\frac{1}{\sqrt{(x-a)(x-b)}}$

Write the minors and cofactors of element of the first column of the following matrices and hence evaluate the determinant in case:
$\text{A}=\begin{vmatrix}2&-1&0&1\\-3&0&1&-2\\1&1&-1&1\\2&-1&5&0 \end{vmatrix}$

Find the equation of the plane passing through the intersection of the planes $2x + 3y - z + 1 = 0$ and $x + y - 2z + 3 = 0$ and perpendicular to the plane $3x - y - 2z - 4 = 0.$

Prove that a necessary and sufficient condition for three vectors $\vec{\text{a}},\ \vec{\text{b}}$ and $\vec{\text{c}}$ to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that $\text{l}\vec{\text{a}}+\text{m}\vec{\text{b}}+\text{n}\vec{\text{c}}=\vec0$.

Solve the following differential equation:
$\text{y dx}+\Big\{\text{x}\log\Big(\frac{\text{y}}{\text{x}}\Big)\Big\}\text{dy}-2\text{x dy}=0$

Solve the following systems of homogeneous linear equations by matrix method: $x + y - z = 0 , x - 2y + z = 0 , 3x + 6y - 5z = 0$

Integrate the function in Exercise:
$\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}$
$\Bigg[\text{Hint:}\frac{1}{\text{x}^{\frac{1}{2}}+\text{x}^{\frac{1}{3}}}=\frac{1}{\text{x}^{\frac{1}{3}}\bigg(1+\text{x}^{\frac{1}{6}}\bigg)},\text{put}\ \text{x}=\text{t}^{6}\Bigg]$

A bag contains 20 tickets, numbered from 1 to 20. Two tickets are drawn without replacement. What is the probability that the first ticket has an even number and the second an odd number.

$\text{Evaluate} \int\limits_0^{2} (x^{2} +x + a) \text{dx as limit of a sum.}$

Solve the following differential equation:
$\text{x}\frac{\text{dy}}{\text{dx}}+2\text{y}=\text{x}\cos\text{x}$