Question

Evaluate the following : `int_(-3)^(3) x^3/(9 - x^2)*dx`

Answer 1

Let I = `int_(-3)^(3) x^3/(9 - x^2)*dx`

Let f(x) = `x^3/(9 - x^2)`

∴ f( –x) = `(-x)^3/(9 - (- x)^2`

= `(-x^3)/(9 - x^2)`

= `-f(x)`

∴ f is an odd function.

∴ `int_-3^3 f(x)*dx = 0, "i.e." int_-3^3 x^3/(9 - x^2)*dx` = 0.