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

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Answer 1 · 2016-06-22T08:39:21

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

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 )#