explain how (y^4)/(1+(y^2)) simplifies to ((y^2) -1 + (1/(1+(y^2))

Question

can't visualise how to simplify this for part of an integration problem

Answer 1

Note that (y² + 1)(y² - 1) = y⁴ - 1  ==>  y⁴ = (y² + 1)(y² - 1) + 1  

 

Substitute this expression in for y⁴ 

 

y⁴ / (1 + y²) = [(y² + 1)(y² - 1) + 1] / (1 + y²) = y² - 1 + [1 / (1 + y²)]    

 

since the (y² + 1) terms cancel in the first fraction