Evaluate the following intregals: 5 (x^2+1)(x+2) dx - Vidyadip

Question

Evaluate the following intregals:
$\int\frac{5}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$

✓

Answer

We have
$\text{I}=\int\frac{5}{(\text{x}^2+1)(\text{x}+2)}\text{ dx}$
Let $\frac{5}{(\text{x}^2+1)(\text{x}+2)}=\frac{\text{A}}{\text{x}+2}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$
$\Rightarrow\frac{5}{(\text{x}^2+1)(\text{x}+2)}=\frac{\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text{x}+2)}{(\text{x}+2)(\text{x}^2+1)}$
$\Rightarrow5=\text{A}(\text{x}^2+1)+\text{Bx}^2+2\text{Bx}+\text{Cx}+2\text{C}$
$\Rightarrow5=(\text{A}+\text{B})\text{x}^2+(2\text{B}+\text{C})\text{x}+(\text{A}+2\text{C})$
Equating coefficient of like terms
A + B = 0 ...(1)
2B + C = 0 ...(2)
A + 2C = 5 ...(3)
Solving (1), (2) and (3), we get
A = 1
B = -1
C = 2
$\therefore\frac{5}{(\text{x}+2)(\text{x}^2+1)}=\frac{1}{\text{x}+2}+\Big(\frac{-\text{x}+2}{\text{x}^2+1}\Big)$
$\Rightarrow\int\frac{5\text{dx}}{(\text{x}+2)(\text{x}^2+1)}+\int\frac{\text{dx}}{\text{x}+2}-\int\frac{\text{x dx}}{\text{x}^2+1}+2\int\frac{\text{dx}}{\text{x}^2+1}$
Let $\text{x}^2+1=\text{t}$
$\Rightarrow2\text{xdx}=\text{dt}$
$\Rightarrow\text{x dx}=\frac{\text{dt}}{2}$
$\therefore\text{I}=\int\frac{\text{dx}}{\text{x}+2}-\frac{1}{2}\int\frac{\text{dt}}{\text{t}}+2\int\frac{\text{dx}}{\text{x}^2+1^2}$
$=\log|\text{x}+2|-\frac12\log|\text{t}|+2\tan^{-1}\text{x}+\text{C}$
$=\log|\text{x}+2|-\frac{1}{2}\log|\text{x}^2+2|+2\tan^{-1}\text{x}+\text{C}$

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

Similar questions

Solve the following system of homogeneous linear equations:

x + y - 2z = 0,

2x + y - 3z = 0,

5x + 4y - 9z = 0

Evaluate the following integrals:
$\int\frac{\cot\text{x}}{\sqrt{\sin\text{x}}}\text{dx}$

Assume that the chances of a patient having a heart attack is $40\%.$ Assuming that a meditation and yoga course reduces the risk of heart attack by $30\%$ and prescription of certain drug reduces its chance by $25\%.$ At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options, the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga. Interpret the result and state which of the above stated methods is more beneficial for the patient.

Evaluate the following integrals:
$\int\limits^{\text{a}}_0\text{x}\sqrt{\frac{\text{a}^2-\text{x}^2}{\text{a}^2+\text{x}^2}}\text{ dx}$

Find the area of the region bounded by the parabolas $y^2 = 4ax$ and $x^2 = 4ay.$

Evaluate: $\int\limits^\frac{3}{2}_{0}|x \cos\pi x| dx$

Show that the curves $2x = y^2$ and $2xy = k$ cut at right angles, if $k^2 = 8.$

Prove that the function
$\text{f}\text{(x)}=\begin{cases}\frac{\text{x}}{|\text{x|+2}\text{x}^2}, &\text{ x}\neq0\\\text{k}, &\text{ x}=0\end{cases}$
remains discontinuous at x = 0, regardless the choice of k.

Solve the following differential equations:$\text{x}\sqrt{1-\text{y}^2}\text{dx}+\text{y}\sqrt{1-\text{x}^2}\text{dy}=0$

$\int\frac{\text{x}}{\sqrt{\text{x}+4}}\text{dx}$