Evaluate the following intregals: 18 (x+2)(x^2+4) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{18}{(\text{x}+2)(\text{x}^2+4)}\text{ dx}$

✓

Answer

Let $\frac{18}{(\text{x}+2)(\text{x}^2+4)}=\frac{\text{A}}{\text{x}+2}+\frac{\text{Bx}+\text{C}}{\text{x}^2+4}$$\Rightarrow18=\text{A}(\text{x}^2+4)+(\text{Bx}+\text{C})(\text{x}+2)$
$18=(\text{A}+\text{B})\text{x}^2+(2\text{B}+\text{C})\text{x}+(4\text{A}+2\text{C})$
Equating similar terms, get,
$\text{A}+\text{B}=0,2\text{B}+\text{C}=0,4\text{A}+2\text{C}=18$
Solving, we get,
$\text{A}=\frac{9}{4},\text{B}=-\frac{9}{4},\text{C}=\frac{9}{2}$
Thus,
$\text{I}=\frac{9}{4}\int\frac{\text{dx}}{\text{x}+2}+\Big(-\frac{9}{4}\Big)\frac{\text{x}}{\text{x}^2+4}\ \text{dx}+\frac{9}{2}\int\frac{\text{dx}}{\text{x}^2+4}$
$\text{I}=\frac{9}{4}\log|\text{x}+2|-\frac{9}{8}\log|\text{x}^2+4|+\frac{9}{4}\tan^{-1}\Big(\frac{\text{x}}{2}\Big)+\text{C}$ $\Big[\because\int\frac{\text{dx}}{\text{x}^2+\text{a}^2}=\frac{1}{\text{a}}\tan^{-1}\frac{\text{x}}{9}\Big]$

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

One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

Evaluate the following integrals as limit of sum:$\int\limits^{4}_{1}\big(\text{x}^2-\text{x}\big)\text{dx}$

Draw a rough sketch of the region bounded by the parabola $y^2 = 4x$ and $x^2 = 4y$ by using methods of integration.

Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{x}-2\sin\text{x}$

Evaluate the following integrals:$\int^\limits2_{-2}|2\text{x}+3|\text{dx}$

If three numbers are added, their sum is $2$. If $2$ times the second number is subtracted from the sum of first and third numbers, we get $8$ and if three times the first number is added to the sum of second and third numbers, we get $4$. Find the numbers using matrices.

If $A$ is a square matrix, using mathematical induction prove that $(A^T)^n = (A^n)^T$ for all $n \in N.$

Show that $\text{f}\text{ (x)}=\begin{cases}\frac{\text{|x}-\text{a}|}{|\text{x}-\text{a}|}, & \text{when} \text{ x}\neq 0\\2, & \text{when}\text{ x} = 0\end{cases}$ is discontinuous at x = a.

Solve the following systems of homogeneous linear equations by matrix method:

$x + y + z = 0$

$x - y - 5z = 0$

$x + 2y + 4z = 0$

If $f(x) = x^3 + 4x^2 - x$, find f(A), where $\text{A}=\begin{bmatrix}0&1&2\\2&-3&0\\1&-1&0\end{bmatrix}$