0-1-x-x-dx-

Question Number 96990 by Ar Brandon last updated on 06/Jun/20

\int_{0}^{1} {(\sqrt{x})}^{\sqrt{x}} dx

Answered by Sourav mridha last updated on 06/Jun/20

\int_{0}^{1} e^{\frac{\sqrt{x}}{2} . l n (x)} d x = \underset{n = 0}{\sum^{\infty}} \frac{1}{2^{n}} \int_{0}^{\infty} \frac{x^{\frac{n}{2}} . {[l n (x)]}^{n}}{n!} d x

n o w l e t l n (x) = - k s o w e g e t

= \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{2^{n} n!} \int_{0}^{\infty} e^{- (\frac{n}{2} + 1) k} . k^{n} d k

= \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{2^{n} Γ (n + 1)} . \frac{Γ (n + 1)}{{(\frac{n}{2} + 1)}^{n + 1}} . [u s i n g L aplace t r a n s f o r m

= 2 \underset{n = 0}{\sum^{\infty}} \frac{{(- 1)}^{n}}{{(n + 2)}^{n + 1}} .

Commented by Ar Brandon last updated on 12/Jun/20

��thanks

Commented by smridha last updated on 12/Jun/20

w e l c o m e

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