Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 37

You need to use the following substitution, such that:
sqrt x = t => (dx)/(2sqrt x) = dt => dx = 2tdt
Replacing the variable yields:
int cos sqrt x dx = int (cos t)*(2tdt)
You need to use the formula of integration by parts, such that:
int udv = uv - int vdu
u = t => du = dt
dv = cos t => v = int cos t dt = sin t
int t*cos t dt = t*sin t - int sin t dt
int t*cos t dt = t*sin t + cos t + C
2int t*cos t dt = 2t*sin t + 2cos t + C
Replacing back the variable sqrt x for t, yields:
int cos sqrt x dx = 2sqrt x*sin(sqrt x) + 2cos(sqrt x)+ C
Hence, evaluating the integral, using substitution and integration by parts yields int cos sqrt x dx = 2sqrt x*sin(sqrt x) + 2cos(sqrt x)+ C.