Question

Fill in the blank : `int_(-9)^9 x^3/(4 - x^2)*dx` = _______

Answer 1

Let I = `int_(-9)^9 x^3/(4 - x^2)*dx`

Let f(x) = `x^3/(4 - x^2)`

∴ f(– x) = `(-x)^2/(4 - (-x)^2`

= `-x^3/(4 - x^2)`

= – f(x)
∴ f(x) is an odd function.

∴ `int_(-9)^9 x^3/(4 - x^2)*dx = 0.      ...[because int_("a")^"a" f(x) = 0, if f(x) "odd function"]`