Question

Evaluate the following integrals : `int_(-9)^9 x^3/(4 - x^2).dx`

Answer 1

Let I = `int_(−9)^9 x^3/(4 − x^2).dx`

We know that, If f(−x) = f(x), f(x) is an even function. If f(−x) = −f(x), f(x) is an odd function.

f(x) = `x^3/(4  –  x^2)`

∴ f(– x) = `(– x)^3/[4  –  ( – x)^2]`

∴ f(– x) = `(−x^3)/(4  –  x^2)`

∴ f(– x) = – f(x)

∴ If f(−x) = −f(x), f(x) is an odd function.

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