Integration by parts- very hard problem?

Question

Use integration by parts to evaluate the definite integral. Integrate from 1 to e.
4 t^2 ln(t) dt

Answer 1

du = 4t^2 dt
v = ln(t)

u = 4/3 t^3
dv = dt/t

∫ vdu = uv - ∫ udv
= 4/3 * t^3 * ln(t) - ∫ 4/3 t^2 dt
= 4/3 * t^3 * ln(t) - 4/9 t^3
= 4/9 t^3 * [3 ln(t) - 1]

Now set the limits to get
4/9 e^3 (2) - 4/9 (-1)
= 4/9 *(2e^3 + 1)

Answer 2

It's not too bad if you approach it correctly:
4 ∫ t² ln t dt
Let U = ln t dV = t²dt
dU = dt/t V = t³/3
So we get
4[ t³ ln t/3 - t³/9]
= 4t³ /9*[ (3 ln t -1) ]
Now plug in your limits:
t = e: 4e³/9*2 = 8e³/9
t = 1: 4/9*(-1) = -4/9
Final answer: (8e³+4)/9.