Question

Prove the following:

`int_(-1)^1 x^17 cos^4 xdx = 0`

Answer 1

Let `I = int_(-1)^1 x^17 cos^4 x " dx" = 0`

Let f(x) = x¹⁷ cos⁴ x

f(-x) = (-x)¹⁷ cos⁴ (-x)

= - x¹⁷ cos⁴ x   ....` [because int_(- a)^a f(x) = 0  "if"  f(- x) = - f]`

∴ I = 0

∴ f (x) is an odd function, hence I = 0.