Prove a simple reduction formula

**harvey123** · February 14th 2013, 04:02 PM

How would I prove the following reduction formula?

∫dx/((x^2)*(sqrt(a^2 - x^2))=-sqrt(a^2 - x^2)/(a^2 x)

**Soroban** · February 14th 2013, 04:50 PM

Hello, harvey123!

This is not a Reduction Formula . . . it is just an integration.

$\int\frac{dx}{x^2\sqrt{a^2-x^2}} \;=\;-\frac{\sqrt{a^2-x^2}}{a^2x} + C$

Let $x \,=\,a\sin\theta \quad\Rightarrow\quad dx \,=\,a\cos\theta\,d\theta$

Substitute: . $\int \frac{a\cos\theta\,d\theta}{a^2\sin^2\theta\!\cdot \!a\cos\theta} \;=\;\frac{1}{a^2}\int\csc^2\theta\,d\theta \;=\; -\frac{1}{a^2}\cot\theta + C$

Back-substitute: . $\sin\theta \,=\,\frac{x}{a} \,=\,\frac{opp}{hyp} \quad\Rightarrow\quad adj \,=\,\sqrt{a^2-x^2}$

. . . . . . . . . . . . . $\cot\theta \,=\,\frac{adj}{opp} \,=\,\frac{\sqrt{a^2-x^2}}{x}$

Answer: . $-\frac{1}{a^2}\cdot\frac{\sqrt{a^2-x^2}}{x}+C \;=\;-\frac{\sqrt{a^2-x^2}}{a^2x} + C$

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