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Question Number 98826 by bramlex last updated on 16/Jun/20

$$Given\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{dx}{a^2+x^2}=\frac{\pi }{2a}$$$$find\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{dx}{{(a^2+x^2)}^3}?$$

Commented by bemath last updated on 16/Jun/20

$$\frac{\mathrm{d}}{\mathrm{da}}\left[\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{\mathrm{dx}}{{\mathrm{a}}^2+{\mathrm{x}}^2}\right]=\frac{\mathrm{d}}{\mathrm{da}}\left[\frac{\pi }{2\mathrm{a}}\right]$$$$-2a\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{\mathrm{dx}}{{({\mathrm{a}}^2+{\mathrm{x}}^2)}^2}=\frac{-\pi }{{2\mathrm{a}}^2}$$$$\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{\mathrm{dx}}{{({\mathrm{a}}^2+{\mathrm{x}}^2)}^2}=\frac{\pi }{{4\mathrm{a}}^3}$$$$\frac{\mathrm{d}}{\mathrm{da}}\left[\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{\mathrm{dx}}{{({\mathrm{a}}^2+{\mathrm{x}}^2)}^2}\right]=\frac{\mathrm{d}}{\mathrm{da}}\left[\frac{\pi }{{4\mathrm{a}}^3}\right]$$$$-4\mathrm{a}\underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{\mathrm{dx}}{{({\mathrm{a}}^2+{\mathrm{x}}^2)}^3}=\frac{-3\pi }{{4\mathrm{a}}^4}$$$$\therefore \underset{0}{\stackrel{\mathrm{\infty }}{\int }}\frac{\mathrm{dx}}{{({\mathrm{a}}^2+{\mathrm{x}}^2)}^3}=\frac{3\pi }{{16\mathrm{a}}^5}◼$$

Commented by bramlex last updated on 16/Jun/20

$$greattt$$

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