# int cos(sqrt(x)) dx First make a substitution and then use integration by parts to evaluate the integral

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.

Approved by eNotes Editorial Team