Question

Evaluate:

`int_(-π/2)^(π/2) (sin^3x)/(1 + cos^2x)dx`

Answer 1

Let f(x) = `(sin^3x)/(1 + cos^2x)`

Then f(– x) = `([sin(-x)]^3)/(1 + [cos(-x)]^2`

= `(-sinx)^3/(1 + (cosx)^2)`

= `- (sin^3x)/(1 + cos^2x)`

= – f(x)

∴ f is an odd function

∴ `int_(-π/2)^(π/2) f(x) dx` = 0

∴ `int_(-π/2)^(π/2) (sin^3x)/(1 + cos^2x)dx` = 0