Gujarat 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$
= $-\int \frac{1}{(x-1)} d x+\frac{1}{2} \int \frac{2 x}{\left(1+x^{2}\right)} d x+\int \frac{1}{\left(1+x^{2}\right)} d x$
= $-\log |x-1|+\frac{1}{2} \log \left|1+x^{2}\right|+\tan ^{-1} x+C$
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