Hi I was just doing some revision on Laplace and this is a easier way to do integration by parts than the ones that I have seen used here

 ∫t^5 e^(-st) dt   

1: define u and v'

u = t^5, v' = e^(-st)

2: then make a table differentiating u and Integrating v' alternating sines down the left

↓d/dt (u)       ∫v'↓

+   t^5          e^(-st)

-   5t^4      -1/s e^(-st)

+  20t^3     1/(s^2) e^(-st)

-   60t^2    -1/(s^3) e^(-st)

+ 120t       1/(s^4) e^(-st)

-  120       -1/(s^5) e^(-st)

+   0          1(/s^6) e^(-st)

3: from the top left multiply the u's by the v's to the right and down one, then for the last one multiply across and leave in integration notation (will be evident why in next example) 

t^5 * (-1)/s e^(-st) + -5t^4 * 1/(s^2) e^(-st) + 20t^3 * (-1)/(s^3) e^(-st) + -60t^2 * 1/(s^4) e^(-st) + 120t * (-1)/(s^5) e^(-st) + -120 * 1/(s^6) e^(-st) + ∫0 * 1/(s^6) e^(-st) dt

4: simplify

e^(-st) (-(t^5)/s - (5t^4)/(s^2) - (20t^3)/(s^3) - (60t^2)/(s^4) - (120t)/(s^5) - 120/(s^6) +0) + C

 ∫sin3t e^(2t) dt  

1: u = e^(2t), v' = sin3t

2:↓d/dt (u)          ∫v'↓

+    e^(2t)          sin3t

-    2e^(2t)      -1/3 cos3t

+   4e^(2t)      -1/9 sin3t

3: ∫sin3t e^(2t) dt = e^(2t) * (-1)/3 cos3t + (- 2)e^(2t) * (-1)/9 sin3t + ∫4e^(2t) * (-1)/9 sin3t dt

4: ∫sin3t e^(2t) dt = e^(2t)(-1/3 cos3t + 2/9 sin3t) -4/9∫sin3t e^(2t) dt

13/9∫sin3t e^(2t) dt = e^(2t)(-1/3 cos3t + 2/9 sin3t) + C

 ∫sin3t e^(2t) dt = 1/13 e^(2t) (2 sin3t - 3 cos3t) +C

thanks