`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...

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.`