My Math Forum Approximation of an integral

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 November 16th, 2010, 09:02 PM #1 Newbie   Joined: Nov 2010 Posts: 2 Thanks: 0 Approximation of an integral My problem is to simplify the following integral: $G=\int_{x=0}^{1}{exp{(tx+kx^{1.5})}}\,dx$ I have a solution, but can't understand it without explanations. The solution is following: $1)k>{\frac{2}{3}t} G_1={\frac{4}{3}}{\frac{\sqrt{\pi}}{k}{t^{1.5}}}ex p({\frac{4}{27}}{t^3}k^{-2})$ $2)k<{\frac{2}{3}t} G_2={\frac{1}{2}}{{\sqrt{\frac{\pi}{a}}}exp{\left( {\frac{b^2-ac}{c}}\right)}{erfc\left({\frac{b}{\sqrt{a}}}\rig ht)$, where $a={\frac{3}{8}}t$, $b={\frac{1}{2}}t-{\frac{3}{4}}k$, $c=k-t$ The second case is clear: we expand the exponential term to the 2nd order Taylor series around u=1, simplify the expression, change variable, set the lower integration linit from 0 to $-\infty$ and get the complimentary error function. But the first case result is not clear at all. Please, help me to understand. Thank you.
 November 17th, 2010, 01:10 PM #2 Global Moderator   Joined: May 2007 Posts: 6,807 Thanks: 717 Re: Approximation of an integral I won't try to answer your question, but I suggest trying a substitution x=u^2.
 November 18th, 2010, 12:05 AM #3 Newbie   Joined: Nov 2010 Posts: 2 Thanks: 0 Re: Approximation of an integral Thanx mathman, I finally solved it without substitution.

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