Question-163587

Question Number 163587 by HongKing last updated on 08/Jan/22

Answered by MJS_new last updated on 08/Jan/22

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

= - \frac{1}{2 (a^{2} + 1)} \int \frac{d x}{x + 1} - \frac{1}{2 (a^{2} - 1)} \int \frac{x + 1}{x^{2} + 1} d x + \frac{a^{2}}{a^{4} - 1} \int \frac{d x}{a^{2} x^{2} + 1} =

= - \frac{1}{2 (a^{2} + 1)} \ln ∣ x + 1 ∣ - \frac{1}{4 (a^{2} - 1)} \ln (x^{2} + 1) - \frac{1}{2 (a^{2} - 1)} \arctan x + \frac{a^{2}}{2 (a^{4} - 1)} \ln (a^{2} x^{2} + 1) + \frac{a}{a^{4} - 1} \arctan (a x) + C

a > 0 : Ω (a) = \frac{a^{2}}{(a^{4} - 1)} \ln a - \frac{(a - 1) π}{4 (a + 1) (a^{2} + 1)} \underset{a \to 1}{and \lim Ω (a) = \frac{1}{4}}

a ⩽ 0 : Ω (a) \notin R

Commented by HongKing last updated on 08/Jan/22

thank you so much my dear Sir cool