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Math Help - How to simplify an integral?

  1. #1
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    How to simplify an integral?

    My problem is to simplify the following integral:
    <br />
G=\int_{x=0}^{1}{exp{(tx+kx^{1.5})}}\,dx<br />
    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}}}\right  ), 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.
    Last edited by nik4214; November 16th 2010 at 08:56 PM.
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