1. Tardigrade
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  4. Let I n =∫ limits01( x ln x ) n dx, if I 4= k ∫ limits01 x 4( ln x )3 dx, then ' k ' is equal to

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Q. Let $I _{ n }=\int\limits_0^1( x \ln x )^{ n } dx$, if $I _4= k \int\limits_0^1 x ^4(\ln x )^3 dx$, then ' $k$ ' is equal to

Integrals

Solution:

$ I_n=\int\limits_0^1(x \ln x)^n d x$
$I_n=\left.\left((\ln x)^n \cdot \frac{x^{n+1}}{n+1}\right)\right|_0 ^1-\frac{n}{n+1} \int\limits_0^1 x^n(\ln x)^{n-1} d x $
$\therefore I_4=\frac{-4}{5} \int\limits_0^1 x^4(\ln x)^3 d x$