Evaluate the integral...?

Question

The integral of p^5 ln p dp.

Answer 1

u can integrate by parts with u= ln p and dv = p^5 or v= p^6/6


u = ln p and v = p^6/6 dv = p^5

int udv = uv - int v du = p^6 ln p/6 - int (p^6/6/p)
= p^6 ln p/ 6 - int p^5/6 = (p^6 lnp)/6 - p^6 + c

Answer 2

∫p^5 ln p dp
∫udv = uv - ∫vdu
Let u = ln(p), dv = p^5 dp, du = dx/x
v = (1/6)p^6
∫udv = uv - ∫vdu = (1/6)p^6ln(p) - ∫(1/6)p^5dx
∫udv = uv - ∫vdu = (1/6)p^6ln(p) - (1/36)p^6 + C
∫p^5 ln p dp = (1/36)p^6(6ln(p) - 1)