help with something dealing with planar curves! How would i do this calc problem?

Question

in this problem t= theta

consider the planar curve given by
x(t)= t ln t , y(t)=sin t 0< t< +infinity

a)calculate x', x'', y', y''

b)find the values 0< t< +infinity where the tangent is horizontal or vertical

c)write down (without calculating) a formula that describes the length of the segment of the curve corresponding to 1< t< +infiniti

Answer 1

x´= 1+ln t x´´= 1/t
y´= cost y´´= -sin t
Horizontal tangent y´=0 so cos t = and t= k*pi with k integer>=0
vertical tangent
x´= 0 1+lnt = 0 so ln t = -1 and t =1/e

L = int (1,infinity) sqrt[(1+ln t)^2+cos^2 t]dt