Evaluate : $\int \frac{2}{(1-x)\left(1+x^{2}\right)}\,dx$

$-\log|1-x|+\dfrac{1}{2}\log\left|1+x^{2}\right|+\tan^{-1}x+C$

CBSE Class 12 · 2012 · Indefinite IntegralsLong Answer

Evaluate :

\int \frac{2}{( 1 - x ) ( 1 + x ^{2} )} d x

Answer: $- lo g ∣1 - x ∣ + \frac{1}{2} lo g 1 + x^{2} + tan^{- 1} x + C$

Resolve into partial fractions with a linear and an irreducible quadratic factor:

\frac{2}{( 1 - x ) ( 1 + x ^{2} )} = \frac{A}{1 - x} + \frac{B x + C}{1 + x ^{2}} .

Comparing coefficients gives

A = B = C = 1

, so

\int \frac{d x}{1 - x} + \int \frac{x + 1}{1 + x ^{2}} d x = - lo g ∣1 - x ∣ + \frac{1}{2} lo g (1 + x^{2}) + tan^{- 1} x + C .

- lo g ∣1 - x ∣ + \frac{1}{2} lo g 1 + x^{2} + tan^{- 1} x + C

CBSE Class 12 Board · 2012 · Subjective