using Plots
using SymPy









@vars x dx
f(x) = sqrt(x^2 + 1)
u = x*log(x)
ex = f(u) * diff(u,x)

integrate(ex, x)

ex =  f(u) * diff(u,x) * dx

u, du = symbols("u, du")
## replace x*log(x) with u and diff(x*log(x),x) * dx with du
ex1 = subs(ex, (x*log(x), u), (dx, du/diff(x*log(x),x)))

ex = integrate(ex1 / du)

subs(ex, (u, x*log(x)))

function usub(ex, let_u_equal)
    u, du, dx = symbols("u, du, dx")
    ex1 = ex * dx
    ex2 = subs(ex1, (dx, du/diff(let_u_equal, x)))
    ex3 = subs(ex2, (let_u_equal, u))
    ex3 / du
end

ex = log(x)/x
usub(ex, log(x))

function udv_parts(ex, let_u_equal)
    u, dv = let_u_equal, ex/let_u_equal
    du = diff(u, x)
    v = integrate(dv)
    [u*v, v*du]			# return two pieces. One for FTC, one to integrate
end

ex = x*sin(x)
uv, vdu = udv_parts(ex, sin(x))



x = symbols("x", positive=true)