Question

What is the antiderivative of 1-cos^2(x)

Answer 1

`(1)/(4)`sin2x +
`(1)/(2)` x + c

Explanation:

using the trigonometric identity

• cos² x =
`(1)/(2)`(cos2x + 1) and

∫cos ax =
`(1)/(a)`sin ax

hence

∫(1 -
`(1)/(2)`(cos2x + 1)dx

= ∫(1 -
`(1)/(2)`cos2x -
`(1)/(2)`)dx

= ∫(
`(1)/(2)` -
`(1)/(2)`cos2x)dx

=
`(1)/(2)`x -
`(1)/(4)`sin2x + c

where c is the constant of integration