March 30th, 2019, 11:36 AM  #1 
Senior Member Joined: Dec 2015 From: iPhone Posts: 486 Thanks: 75  Evaluate integral
How to evaluate the integral ? $\displaystyle \int_{0}^{1} x^{n} e^x dx \; \; $ , $\displaystyle n\in \mathbb{N}$. 
March 30th, 2019, 02:17 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,732 Thanks: 689 
Repeated integrate by parts. $\int_0^1x^ne^xdx=x^ne^x]_0^1n\int_0^1x^{n1}e^xdx$ The first term $=e$. Keep going till you hit 0.

March 31st, 2019, 02:42 PM  #3 
Senior Member Joined: Dec 2015 From: iPhone Posts: 486 Thanks: 75 
Got the answer like : $\displaystyle (1)^{n1}+e\sum_{j=0}^{n} (1)^{nj} \frac{n! }{j! }$.


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