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Question Number 211696 by mnjuly1970 last updated on 16/Sep/24

$$$$$${\underset{0}{\int }}^1\frac{1}{{(2+2x+x^2)}^3}dx=?$$$$\underbrace{\underset{\stackrel{\mathrm{Improper}\mathrm{integral}}{}}{}}$$$$---------$$

Answered by Ghisom last updated on 16/Sep/24

$$\int \frac{dx}{{(x^2+2x+2)}^3}=$$$$\left[{\mathrm{Ostrogradski}}^{\prime }\mathrm{s}\mathrm{Method}\right]$$$$=\frac{(x+1)(3x^2+6x+8)}{8{(x^2+2x+2)}^2}+\frac{3}{8}\int \frac{dx}{x^2+2x+2}=$$$$=\frac{(x+1)(3x^2+6x+8)}{8{(x^2+2x+2)}^2}+\frac{3}{8}\arctan (x+1)+C$$$$\Rightarrow$$$$\underset{0}{\stackrel{1}{\int }}\frac{dx}{{(x^2+2x+2)}^3}=-\frac{2}{25}+\arctan \frac{1}{3}$$

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