# Difficult integral

1. Sep 24, 2010

### matteo86bo

I don't know where to start with this integral ...

$$\int_0^t\frac{e^{-x/h}(a+b(x-c))xdx}{(a+b(x-c)-1)}$$

can you give me a hand?

2. Sep 24, 2010

### arkajad

This integral does not integrate in terms of elementary functions. Are you sure the formula is correct? Or, perhaps, you know about "exponential integral" Ei?

3. Sep 24, 2010

### jackmell

Learn how to encapsulate and generalize:

$$\int_0^t\frac{e^{-x/h}(a+b(x-c))xdx}{(a+b(x-c)-1)}=\int\frac{e^{-x/h}(k+x)x}{r+bx}=k\int\frac{x e^{-x/h}}{r+bx}dx+\int\frac{x^2 e^{-x/h}}{r+bx}dx$$

Now, suppose I tell you the function:

$$Ei(z)=-\int_{-z}^{\infty}\frac{e^{-t}}{t}dt$$

can be treated just like any other function like sin and cosine. For example, what happens when you differentiate Ei(z)? Knowing that, can you then express the antiderivative of your integral in terms of some expression which contains Ei(z) where z is some combination of the variables and constants in your integrand?

4. Sep 25, 2010

### matteo86bo

Sorry but I've just asked if there exists an analytical solution ...
I've been dealing with these kind of integrals in my thesis and I always have to solve them numerically ...
thanks for the advices!

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