Question

Evaluate the following integrals:

\[\int\frac{x^2}{\left( a^2 - x^2 \right)^{3/2}}dx\]

Answer 1

\[\text{Let I }= \int\frac{x^2}{\left( a^2 - x^2 \right)^ {3/2}}dx\]

\[ \text{Let x }= a \cos\theta\]

`" 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 `