Integration : Mapping Smoothly (-inf, 2] to [0.1,0.9]

[Hepth]

I have an Integral that is convergent over the range (-inf, Lambda) where 0< Lambda < 1.

I need to change variables to move this to (0.1, 0.9) in such a way that I do not introduce any poor behavior, such as asymptotes or discontinuities as it needs to be well behaved.

Is there a standard practice for this, like when mapping to unit cube/square?

The integral is like :

$\int_{-\infty}^{\Lambda} e^{3 x} (\Lambda - x)^3 P[x] dx$

where P[x] is some generic polynomial or unknown smooth function.

Thanks for your help!

-Hepth

EDIT :
I guess I can just use something like
$\Lambda + \frac{(t-\text{tp})}{t-\text{tm}}$

 

[Svein]

Try $u=\frac{0.8}{\Lambda x}+0.1$.