Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIntegrals3 Marks
Question
Integrate the rational function $\frac{2}{(1-x)\left(1+x^{2}\right)}$
✓
Answer
Let $\frac{2}{(1-x)\left(1+x^{2}\right)}=\frac{A}{(1-x)}+\frac{Bx + C}{\left(1+x^{2}\right)}$
$\Rightarrow 2 = A(1 + x^2) + (Bx + C)(1 - x)$
$\Rightarrow 2 = A + Ax^2 + Bx - Bx^2 + C - Cx$
On comparing the coefficients of $x^2, x$ and constant term, we get,
$A - B = 0$
$B - C = 0$
$A + C = 2$
On solving these equations, we get,
$A = 1, B =1$ and $C = 1$
Thus,
$\frac{2}{(1-x)\left(1+x^{2}\right)}=\frac{1}{(1-x)}+\frac{x+1}{\left(1+x^{2}\right)}$
$\Rightarrow$$\int \frac{2}{(1-x)\left(1+x^{2}\right)} d x=\int \frac{1}{(1-x)} d x+\int \frac{x}{\left(1+x^{2}\right)} d x+\int \frac{1}{\left(1+x^{2}\right)} d x$
$=-\int \frac{1}{(x-1)} d x+\frac{1}{2} \int \frac{2 x}{\left(1+x^{2}\right)} d x+\int \frac{1}{\left(1+x^{2}\right)} d x$
$= -\log |x-1|+\frac{1}{2} \log \left|1+x^{2}\right|+\tan ^{-1} x+C$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.