self doubt

Question

How to solve these questions

$(1)$ $I=\int_{0}^{1}(xlogx)^{4}dx$

$(2)$  $I=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}e^{\frac{-x^{2}}{8}}dx$

$(3)$  $I=\int_{0}^{\infty}x^{\frac{1}{4}}.e^{-\sqrt{x}}dx$

Comments

$(1)$ $I=\frac{4!}{5^{5}}$

$(2)$ $I=1$

$(3)$ $I=\frac{3}{2}\times\sqrt{\pi}$

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3rd one easy

will be $e^{-\sqrt{\pi }}\left [ \frac{x^{5/4}}{5/4} \right ]$ now put limit

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@srestha ma'am see the question again I edit 

i have problem with $(1)?$

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1st one can be done using integration by parts though it's very lengthy.Taking the 1st function as $log^{4}x$ and 2nd function as x^4.Getting the ans as 24/5^5

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yes, it is very lengthy.

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first one

Take log x = t
dx = x dt
e^t = x

then solve using integration by parts

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how to change limit ?

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In 2) put $x^{2}=z$

$2x.dx=dz$

$dx=dz/2\sqrt{z}$

then put partial integration

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for Q2 it can be solved by normal distribution function here sigma =2 MU=0 put in the equation answer will be 1

for Q3 it will be solved by gamma  Function method https://study.com/academy/lesson/gamma-function-properties-examples.html

https://www.youtube.com/watch?v=tp0HiqJPh_E

for Q1 my approach is lengthy unable to get the answer   

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thanks