Question

int_(2)^(oo)(1)/((x-1)(x^(2)+1))*dx`

Answer 1

Answer:

$\huge \: Question$

$\sf \displaystyle \pink{ \int _{2}^{ \infty } \frac{1}{(x - 1)( {x}^{2} + 1)} \: dx}$

To evaluate the improper integral , by definition , rewrite it using a limit and a definite integral.

$\displaystyle \pink{ \: \lim_{a \to + \infty }( \int_{2}^{a} \frac{1}{(x - 1) \times ( {x}^{2} + 1) } dx}$

$\small \pink{ \lim_{a \to \: \infty }( \frac{1}{2}( |a - 1| - \frac{1}{4} \times ln( {a}^{2} + 1) + \frac{ - \arctan(a) + \arctan(2)}{2} + \frac{1}{4} \times ln(5) }$

Evaluate the limit.

$\displaystyle \pink{\: \frac{ - \pi \: + 2 \arctan(2)}{4} + \frac{1}{4} \times ln(5) }$