if f'(x) = ( x*cosx + 7) /x and f(0) = 5. find f(x).

f'(x) = (x*cosx + 7) /x

Let
us simplify the function.

f'(x) = xcosx/x +
7/x

       = cosx +
7/x

==> f'(x) = cosx +
7/x

Now we know that f(x) = integral
f'(x).

==> f(x) = intg f'(x)
dx

             = intg (cosx + 7/x)
dx

             = intg cosx dx  + intg 7/x
dx

             = sinx + 7*ln x +
C

==>f(x) = sinx  + 7*lnx +
C

f(0) is not a valid point
because lnx is not defined at x= 0.