Question

Evaluate the following : `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`

Answer 1

Let I = `int_((-pi)/4)^(pi/4) x^3 sin^4x*dx`

Let f(x) = `x^3 sin^4x`

∴ f( –x) = `(-x)^3 sin^4(- x)`

= `-x^3sin^4x`

= `-f(x)`

∴ f is an odd function.

∴ `int_((-pi)/4)^(pi/4) f(x)*dx = 0, "i.e." int_((-pi)/4)^(pi/4) x^3 sin^4x*dx` = 0.