Thread: How to simplify an integral?

1. How to simplify 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) $k<{\frac{2}{3}t}$
$G_2={\frac{1}{2}}{{\sqrt{\frac{\pi}{a}}}exp{\left( {\frac{b^2-ac}{c}}\right)erfc\left({\frac{b}{\sqrt{a}}}\right )$, 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.