Question

What is the integral of `(cos(x^(1/2)))`?

Answer 1

`2 (sqrt(x) sin sqrt(x) + cos sqrt(x) + C )`

Explanation:

obvious sub `q = sqrt(x), dq = (dx)/(2 sqrt(x)) \implies dx = 2 q \ dq`

so the integral is `2 \int \ q cos q \ dq`

integration by parts using form `int u v' = uv - int u' v`

here `u = q, u' = 1` and `v' = cos q, v = sin q`

so we have

`2 (q sin q - int sin q \ dq )`

`= 2 (q sin q + cos q + C )`

`= 2 (sqrt(x) sin sqrt(x) + cos sqrt(x) + C )`