Question

integration 2/(1-x)(1+x^2)​

Answer 1

Step-by-step explanation:

We have,

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

$= \int \frac{(1 + {x}^{2} ) + (1 - x)(1 + x)dx}{(1 - x)(1 + {x}^{2} ) }$

$= \int \frac{dx}{(1 - x)} + \int \frac{1 + x}{1 + {x}^{2} } dx$

$= - ln(1 - x) + \int \frac{dx}{ {x}^{2} + 1 } + \frac{1}{2} \int \frac{2x}{1 + {x}^{2} } dx$

$= - ln(1 - x) + \tan ^{- 1} (x) + \frac{1}{2} ln(1 + {x}^{2} ) + c$

$= - ln(1 - x) + \tan ^{- 1} (x) + ln \sqrt{1 + {x}^{2} } + c$