Solve Difficult Integral - Get Help Now!

[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?

 

[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?

 

[jackmell]

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?

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?

 

[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!