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

Question

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

✓

Answer

Let $\frac{1}{(\text{x}^2+1)(\text{x}^2+4)}=\frac{\text{Ax}+\text{B}}{(\text{x}^2+1)}+\frac{\text{Cx}+\text{D}}{\text{x}^2+4}$
$\Rightarrow1=(\text{Ax}+\text{B})(\text{x}^2+4)+(\text{Cx}+\text{D})(\text{x}^2+1)$
$=(\text{A}+\text{C})\text{x}^3+(\text{B}+\text{D})\text{x}^2+(4\text{A}+\text{C})\text{x}+4\text{B}+\text{D}$
Equating similar terms, we get,
A + C = 0, B + D = 0, 4A + C = 0, 4B + D = 1
Solving, we get, $\text{A}=0,\text{B}=\frac{1}{3},\text{C}=0,\text{D}=-\frac{1}{3}$
Thus,
$\text{I}=\int\frac{\frac{1}{3}\text{dx}}{(\text{x}^2+1)}-\int\frac{\frac{1}{3}\text{dx}}{(\text{x}^2+4)}$
$=\frac{1}{3}\tan^{-1}\text{x}-\frac{1}{6}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$ $\big[\because\int\frac{\text{dx}}{\text{x}^2+\text{a}^2}=\frac{1}{\text{a}}\tan^{-1}\frac{\text{x}}{\text{a}}\big]$
$\therefore\text{I}=\frac{1}{3}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$

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 INTEGRALS→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 the intervals in which $f(x) = (x + 2)e^{-x}$ is increasing or decreasing.

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.
$f(x) = x^3 - 2x^2 - x + 3$ on $[0, 1]$

Find the coordinates of the point where the line $\frac{\text{x}-2}{3}=\frac{\text{y}+1}{4}=\frac{\text{z}-2}{2}$ intersectscts the plane x - y + z - 5 = 0. Also, find the angle between the line and the plane.

Solve the following differential equation: $\frac{\text{dy}}{\text{dx}} = \frac{\text{x}(\text{2y - x})}{\text{x}(\text{2y+ x)}}\text{If, y = 1 when x = 1} $

Solve the following systems of linear equations by cramer's rule:

2x - 3z + w = 1,

x - y + 2w = 1,

-3y + z + w = 1,

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

Minimize Z = 3x + 2y subject to the constraints:
$x + y \geq 8$
$3 x + 5 y \leq 15$
$x \geq 0 , \ y \geq 0$

A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die.

A manufacturer can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.

	A	B
Grinding	1	2
Turning	3	1
Assembling	6	3
Testing	5	4

The available capacities of these operations in hours for the given time period are: grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs 20 for each unit of A and Rs 30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.

Differentiate the following functions with respect to x:
$\sqrt{\tan^{-1}\big(\frac{\text{x}}{2}\big)}$