Integration of limit 0 to 1 x^3(logx)^3 dx.

Question

\(\int\limits_0^1x^3(log x)^3dx\)

Answer 1

Use integration by parts:- integration from 0 to 1 x³ (log x)³ dx

Answer 2

I = \(\int\limits_0^1x^3(log x)^3dx\)

let log x = -t ⇒ x = e^-t

1/x dx = -dt

Limits converts info from t = -log (0) = -(-\(\infty\)) = \(\infty\)

to f = -log 1 = -0 = 0

∴ I = \(\int\limits_{\infty}^0e^{-4t}(-t)^3dt\) = \(-\int\limits_0^{\infty}t^3e^{-4t}dt\) 

= \(\frac{\Gamma(4)}{4^4}\) ( ∵ \(\int\limits_0^{\infty}\)x^n-1e^-axdx = \(\frac{\Gamma(n)}{a^n}\), n = 4)

 ∴ \(\int\limits_0^1\)(x log x)³dx = 3/128