Integration problem

Question

Solve step by step without using partial fractions if possible.

 

Answer 1

∫ x³ dx / (a² - x² )

= ∫ x² x dx / (a² - x² )

= ∫ - (a² - x² - a²) x dx / (a² - x² )

= [∫ - (a² - x²)xdx/(a² - x² )] + [∫a² x dx / (a² - x² )]

=[∫ - xdx] + [∫a² x dx / (a² - x² )]

= - [x²/2] - (a²/2)ln(a² - x² ) + C