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प्रश्न
उत्तर
\[\text{Let I }= \int\frac{x^2}{\left( a^2 - x^2 \right)^ {3/2}}dx\]
\[ \text{Let x }= a \cos\theta\]
`" On differentiating both sides, we get " `
`dx = - a sin θ dθ `
` ∴ I = ∫ {a^2cos^2θ} /( a^2 - a^2cos^2 θ )^{3 /2} ×- a sin θ dθ `
` = - ∫ {a^3 cos^2θ sinθ } /( a^3 (1 - cos^2 θ )^{3 /2} dθ`
` = - ∫ {cos^2θ sin θ } /( sin^3 θ ) dθ`
` = - ∫ cot^2 θ dθ`
\[ = - \int\left( \ cose c^2 \theta - 1 \right) d\theta\]
\[ = - \left( - \cot\theta - \theta \right) + c\]
\[ = \cot\theta + \theta + c\]
\[ = \cot\left( \cos^{- 1} \frac{x}{a} \right) + \cos^{- 1} \frac{x}{a} + c\]
\[ = \cot\left( \cot^{- 1} \frac{x}{\sqrt{a^2 - x^2}} \right) + \cos^{- 1} \frac{x}{a} + c\]
\[ = \frac{x}{\sqrt{a^2 - x^2}} + \cos^{- 1} \frac{x}{a} + c\]
` Hence, ∫ x^2 /( a^2 - x^2) ^{3/2}dx = x / \sqrt {a^2-x^2} + cos ^-1 x/a + c `
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