Question

Find : $\int \frac{x^{2}}{\sqrt{a^{6}-x^{6}}}dx$​

Answer 1

Heya!!

We know that integration of 1/√ ( a² - x² ) dx is

Sin-¹ x/a

______________________________

Integration of

x² dx
___________
√{ (a³)² - ( x³ )² }

=>

Put x³ = z

Differentiate both sides w.r.t x we get

3x² dx = dz

dx = dz/3x²

=>

integration of

x² dz
_____________
3x² { √ { (a³ )² - z² } }

=>

integration of

dz
_____________
3 { √{ (a³ )² - z² } }

=

1/3 { Sin-¹ ( z / a³ )

=

1/3 { Sin-¹ ( x³ / a³ ) + A

Where A is called constant of integration.