Download App

Indefinite Integrals — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIndefinite Integrals4 Marks
Question
Evaluate the following intregals:
$\int\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}\ \text{dx}$
✓

Answer

Consider the integrals
$\text{I}=\int\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}\ \text{dx}$
Now let us separate the fraction $\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}$
Through particle fractions.
$\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{\text{A}}{\text{x}-1}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$
$\Rightarrow\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text {x}-1)}{(\text{x}^2+1)(\text{x}-1)}$
$\Rightarrow\text{x}=\text{A}(\text{x}^2+1)+(\text{Bx}+\text{C})(\text{x}-1)$
Compairing the coefficient, we have,
A + B = 0, -B + C = 1 and A - C = 0
Solving the equations we, get,
$\Rightarrow\text{A}=\frac{1}{2},\text{B}=-\frac{1}{2}\text{ and }\text{C}=\frac{1}{2}$
$\Rightarrow\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{\text{A}}{\text{x}-1}+\frac{\text{Bx}+\text{C}}{\text{x}^2+1}$
$\Rightarrow\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{1}{2}\times\frac{1}{\text{x}-1}-\frac{1}{2}\times\frac{\text{x}-1}{\text{X}^2+1}$
$\Rightarrow\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}=\frac{1}{2(\text{x}-1)}-\frac{\text{x}}{2(\text{x}^2+1)}+\frac{1}{2(\text{x}^2+1)}$
Thus, we have,
$=\int\frac{\text{x}}{(\text{x}^2+1)(\text{x}-1)}\ \text{dx}$
$=\int\Big[\frac{1}{2(\text{x}-1)}-\frac{\text{x}}{2(\text{x}^2+1)}+\frac{1}{2(\text{x}^2+1)}\Big]\text{dx}$
$=\int\frac{\text{dx}}{2(\text{x}-1)}-\int\frac{\text{xdx}}{2(\text{x}^2+1)}+\int\frac{\text{dx}}{2(\text{x}^2+1)}$
$=\int\frac{1}{2}\frac{\text{dx}}{(\text{x}-1)}-\int\frac{1}{2}\frac{\text{xdx}}{(\text{x}^2+1)}+\int\frac{1}{2}\frac{\text{dx}}{(\text{x}^2+1)}$
$=\frac{1}{2}\int\frac{\text{dx}}{(\text{x}-1)}-\frac{1}{2}\times\frac{1}{2}\int\frac{2\text{xdx}}{(\text{x}^2+1)}+\frac{1}{2}\int\frac{\text{dx}}{(\text{x}^2+1)}$
$=\frac{1}{2}\log|\text{x}-1|-\frac{1}{4}\log(\text{x}^2+1)+\frac{1}{2}\tan^{-1}\text{x}+\text{C}$

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 "4 Marks" 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

If the product of the distances of the point (1, 1, 1) from the origin and the plane x - y + z + λ = 0 be 5, find the value of λ.
→
Evaluate the following integrals:
$\int\limits^1_{-1}\log\Big(\frac{2-\text{x}}{2+\text{x}}\Big)\text{dx}$
→
Solve the following system of equations by matrix method:
6x - 12y + 25z = 4
4x + 15y - 20z = 3
2x + 18y + 15z = 10
→
$\text{if } \vec{\text{a}} = 2\hat{\text{i}} + \hat{\text{j}} - \hat{\text{k}}, \vec{\text{b}} = 4\hat{\text{i}} - 7\hat{\text{j}} + \hat{\text{k}}, \text{find a vector } \vec{\text{c}} \text{ such that } \vec{\text{a}} \times \vec{\text{c}} \text{ and } \vec{\text{a }} . \vec{\text{c}} = 6.$
→
Find the area bounded by the parabola y2 = 4x and the line y = 2x - 4:
By using horizontal strips.
→
Radium decomposes at a rate proportional to the quantity of radium present. It is found that in 25 years, approximately 1.1% of a certain quantity of radium has decomposed. Determine approximately how long it will take for one-half of the original amount of radium to decompose?
→
Find the general solution of $(\text{x}+2\text{y}^3)\frac{\text{dy}}{\text{dx}}=\text{y}.$
→
Find the general solution $ x \frac { d y } { d x } + y - x + x y \cot x = 0 \ (x \neq 0)$.
→
verify that $\text{y}=\log(\text{x}+\sqrt{\text{x}^2+\text{a}^2})^2$ is a solution of the differential equation $(\text{a}^2+\text{x}^2)\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{x}\frac{\text{dy}}{\text{dx}}=0$
→
Fine: $\int\frac{3\text{x}+5}{\text{x}^2+3\text{x}-18}\text{dx}.$
→