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Question Number 203884 by Mathspace last updated on 31/Jan/24

$$find{\int }_0^{\mathrm{\infty }}\frac{x^3}{{(1+x)}^4{(x+2)}^5}dx$$

Answered by Frix last updated on 01/Feb/24

$$\mathrm{Use}{\mathrm{Ostrogradski}}^{\prime }\mathrm{s}\mathrm{Method}\mathrm{to}\mathrm{get}$$$$\int \frac{x^3}{{(x+1)}^4{(x+2)}^5}dx=$$$$=\frac{288x^6+2736x^5+10608x^4+21432x^3+23745x^2+13666x+3194}{3{(x+1)}^3{(x+2)}^4}+96\int \frac{dx}{(x+1)(x+2)}$$$$\Rightarrow \mathrm{answer}\mathrm{is}$$$$96\ln 2-\frac{1597}{24}$$

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