anonymous
  • anonymous
I need help with an Integral (Substitution).
Mathematics
katieb
  • katieb
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anonymous
  • anonymous
\[\int\limits_{0}^{100}e^\sqrt{x}dx=?\] I am supposed to solve this using the method of substitution. I have no clue how the hell this would help in that weird case.
hartnn
  • hartnn
u know integration by parts ?
across
  • across
Let\[ u=\sqrt x.\tag{1}\]Then,\[ du=\frac1{2\sqrt x}dx\Longrightarrow dx=2\sqrt xdu=2udu.\tag{2} \]It follows from \((1)\) that\[ x=0\Longrightarrow u=0\text{, and}\\ x=100\Longrightarrow u=10. \]So, from \((1)\), \((2)\) and the new limits, we have that\[ 2\int_0^{10}ue^u\,du. \]

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anonymous
  • anonymous
@hartnn Yes, that would be my first impulse but I should use the substitution method.
hartnn
  • hartnn
as shown by @across
anonymous
  • anonymous
Ok, and after that I have to integrate by parts do integrate the \[ue^udu\] part, right?
hartnn
  • hartnn
yes.
across
  • across
Yes, or you could use the (not so) common knowledge that\[ \int xe^x\,dx=e^x(x-1). \]
anonymous
  • anonymous
^^ I thought that I saw that Integral before, but not today it seems. Thx for your help.
hartnn
  • hartnn
more general formula : \(\large \int e^x(f(x)+f'(x))dx=e^xf(x)+c\)

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