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Question Number 104505 by mathmax by abdo last updated on 22/Jul/20

calculate \int_{0}^{\infty} \frac{dx}{{(x + 1)}^{2} {(x + 2)}^{2} {(x + 3)}^{2}}

Answered by OlafThorendsen last updated on 22/Jul/20

R (x) = \frac{1}{{(x + 1)}^{2} {(x + 2)}^{2} {(x + 3)}^{2}}

R (x) = \frac{1}{4 {(x + 1)}^{2}} + \frac{1}{{(x + 2)}^{2}} + \frac{1}{4 {(x + 3)}^{2}}

- \frac{3}{4 (x + 1)} + \frac{3}{4 (x + 3)}

\int_{0}^{\infty} R (x) dx =

{[- \frac{1}{4 (x + 1)} - \frac{1}{x + 2} - \frac{1}{4 (x + 3)} - \frac{3}{4} \ln ∣ \frac{x + 1}{x + 3} ∣]}_{0}^{\infty}

= \frac{1}{4} + \frac{1}{2} + \frac{1}{12} + \frac{3}{4} \ln \frac{1}{3}

= \frac{5}{6} - \frac{3}{4} \ln 3