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

Question

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

✓

Answer

We have,
$\text{I}=\int\frac{\text{dx}}{\text{x}(\text{x}^4-1)}$
$=\int\frac{\text{x}^3\text{dx}}{\text{x}^4(\text{x}^4-1)}$
Putting $\text{x}^4=\text{t}$
$\Rightarrow4\text{x}^3\text{dx}=\text{dt}$
$\Rightarrow\text{x}^3\text{dx}=\frac{\text{dt}}{4}$
$\therefore\text{I}=\frac{1}{4}\int\frac{\text{dt}}{\text{t}(\text{t}-1)}$
Let $\frac{1}{\text{t}(\text{t}-1)}=\frac{\text{A}}{\text{t}}+\frac{\text{B}}{\text{t}-1}$
$\rightarrow\frac{1}{\text{t}(\text{t}-1)}=\frac{\text{A}(\text{t}-1)+\text{B}\text{t}}{(\text{t}-1)}$
$\Rightarrow1=\text{A}(\text{t}-1)+\text{Bt}$
Putting t - 1 = 0
⇒ t = 1
$\therefore$ 1 = A × 0 + B (1)
⇒ B = 1
Putting t = 0
$\therefore$ 1 = A (0 - 1) + B × 0
⇒ A = -1
$\therefore$ $\text{I}=-\frac{1}{4}\int\frac{\text{dt}}{\text{t}}+\frac{1}{4}\int\frac{\text{dt}}{\text{t}-1}$
$=-\frac{1}{4}\log|\text{t}|+\frac{1}{4}\log|\text{t}-1|+\text{C}$
$=\frac{1}{4}\log\Big|\frac{\text{t}-1}{\text{t}}\Big|+\text{C}$
$=\frac{1}{4}\log\Big|\frac{\text{x}^2-1}{\text{x}^4}\Big|+\text{C}$

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 "5 Marks Questions" 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

Solve the following differential equation:
$(\text{x}+\text{y}+1)\frac{\text{dy}}{\text{dx}} = 1$

Solve the following L.P.P. graphically:
Maximise $Z = 20x + 10y$
Subject to the following constraints $x + 2y \leq 28,$
$3x + y \leq 24,$
$x \geq 2,$
$x, y \geq 0$

Show that the following system of linear equation is inconsistent:
4x − 5y − 2z = 2
5x − 4y + 2z = −2
2x + 2y + 8z = −1

For the following differntial equations verify that the accompanying function is a solution:

Differential equation	Function
$\text{x}\frac{\text{dy}}{\text{dx}}+\text{y}=\text{y}^2$	$\text{y}=\frac{\text{a}}{\text{x}+\text{a}}$

Find the angle between the vectors whose direction cosines are proportional to 2, 3, -6 and 3, -4, 5.

Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = x(x + 4)^2$ on $[0,4]$

Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point $(x, y)$ is equal to the sum of the coordinates of the point.

In a game, a man wins ₹ 5 for getting a number greater than 4 and loses ₹ 1 otherwise, when a fair die is thrown. The man decided to throw a die three but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses.

$\text{If A} = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}, $ verify that $\text{A}^{2} - \text{4A - 5I = 0}$

If $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ are non-coplanar vectors, prove that the point having the following position vectors is collinear:$\vec{\text{a}},\ \vec{\text{b}},\ 3\vec{\text{a}}-2\vec{\text{b}}$