Download App

INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS5 Marks
Question
Integrate the rational function in exercise:
$\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}$
✓

Answer

$\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{\text{Ax}+\text{B}}{\text{x}^2+1}+\frac{\text{C}}{\text{x}-1}\dots(\text{i)}$
$\Rightarrow x = (Ax + B) (x - 1) + C(x^2 + 1)$
$\Rightarrow x = Ax^2 - Ax + Bx - B + Cx^2 + C$
Comparing coefficients of$ x^2, A + C = 0 ....(ii)$
Comparing coefficients of $x, –A + B = 1 ....(iii)$
Comparing constant terms, $–B + C = 0 ….(iv)$
Solving eq. (ii), (iii) and (iv), we get
$\text{A}=\frac{-1}{2},\ \text{B}=\frac{1}{2} \ \text{and} \ \text{C}=\frac{1}{2}$
Putting the values of A, B and C in eq. (i),
$\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{\frac{-1}{2}\text{x}+\frac{1}{2}}{\text{x}^2+1}+\frac{\frac{1}{2}}{\text{x}-1}$
$\Rightarrow \ \frac{\text{x}}{(\text{x}^2+1)}=\frac{-1}{2}.\frac{\text{x}}{\text{x}^2+1}+\frac{1}{2}.\frac{1}{\text{x}^2+1}+\frac{1}{2}.\frac{1}{\text{x}-1}=\frac{-1}{4}.\frac{2\text{x}}{\text{x}^2+1}+\frac{1}{2}.\frac{1}{\text{x}^2+1}+\frac{1}{2}.\frac{1}{\text{x}-1}$
$\Rightarrow \ \frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}\text{dx}=\frac{-1}{4}\int\frac{2\text{x}}{\text{x}^2+1}\text{dx}+\frac{1}{2}\int\frac{1}{\text{x}^2+1}\text{dx}+\frac{1}{2}\int\frac{1}{\text{x}-1}\text{dx}$
$​​​​​​\Rightarrow \ \frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}\text{dx}=\frac{-1}{4}\text{log}|\text{x}^2+1|+\frac{1}{2}\tan^{-1}\text{x}+\frac{1}{2}\text{log}|\text{x}-1|+\text{c}$
$​​​​​​\Rightarrow \ \frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{-1}{4}\text{log}(\text{x}^2+1)+\frac{1}{2}\tan^{-1}\text{x}+\frac{1}{2}\text{log}|\text{x}-1|+\text{c}$

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

Minimise Z = x + 2y
subject to $2\text{x}+\text{y}\geq3,\ \text{x}+2\text{y}\geq6,\ \text{x},\ \text{y}\geq0.$
Show that the minimum of Z occurs at more than two points.
→
Prove that $\int\limits_{0}^{\text{a}}\text{f(x)dx}=\int\limits_{0}^{\text{a}}\text{f (a - x) dx.}$
Hence, evaluate $\int\limits_{0}^{\pi/2}\frac{\text{dx}}{\text{1 + tan x}}$.
→
If $\vec{\text{a}}=2\hat{\text{i}}-3\hat{\text{j}}+\hat{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+\hat{\text{k}},\vec{\text{c}}=2\hat{\text{j}}-\hat{\text{k}}$are three vectors, find the aera of the parallelogram having diagonals $\big(\vec{\text{a}}+\vec{\text{b}}\big)$ and $\big(\vec{\text{b}}+\vec{\text{c}}\big).$
→
Evaluate the following integrals:
$\int(2\text{x}-5)\sqrt{\text{x}^2-4\text{x}+3}\text{dx}$
→
find the area of the region included between the parabola $y^2 = x$ and the line $x + y = 2$.
→
Find the length and the foot of perpendicular from the poin $\Big(1,\frac{3}{2},2\Big)$ to the plane 2x - 2y + 4z + 5 = 0.
→
A die is thrown thrice. Find the probability of getting an odd number at least once.
→
Evaluate:
$\begin{vmatrix}1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}$
→
A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
A = the first throw results in head,
B = the last throw results in tail.
→
Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,$f'(x) = (x - 1)(x + 2)^2$
→