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\(\int\limits_0^1x^3(log x)^3dx\) |
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Answer» Use integration by parts:- integration from 0 to 1 x3 (log x)3 dx 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 = \(\int\limits_{\infty}^0e^{-4t}(-t)^3dt\) = \(-\int\limits_0^{\infty}t^3e^{-4t}dt\) = \(\frac{\Gamma(4)}{4^4}\) ( ∵ \(\int\limits_0^{\infty}\)xn-1e-axdx = \(\frac{\Gamma(n)}{a^n}\), n = 4) ∴ \(\int\limits_0^1\)(x log x)3dx = 3/128 |
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