# Integral of Ei(x)

Here I talked about special functions, and at the end of the post I said I would explain how to integrate these special functions. So here I am, showing you how to integrate the function Ei(x). If you gave it a try, let me know in the comments if you used this approach!

$\int&space;\text{Ei}(x)\,&space;dx$

We said that Ei(x) is equal to the integral of (e^x)/x, which means that the derivative of Ei(x) is (e^x)/x. What we need to figure out is what integration technique will actually be helpful. Honestly, u-substitution is not going to do much for us, as you will see:

$u=\text{Ei}(x)\;\;\;\;\;du=\frac{e^x}{x}dx\rightarrow&space;dx=\frac{x}{e^x}du$ $\int&space;u\frac{x}{e^x}\,&space;du$

We don’t want any x, since we are in the u-world, but we notice that u=Ei(x), therefore

$x=\text{Ei}^{-1}(u)$

but what does that mean?? We are obviously not going that far. This means we have to try with integration by parts. Ei(x) is equal to Ei(x)*1. Since the integral of Ei(x) is what we want to find, we will let u be Ei(x) so that we can differentiate it and let dv be 1.

$u=\text{Ei}(x)\:&space;\:&space;\:&space;\:&space;\:&space;dv=1\,&space;dx$ $du=\frac{e^x}{x}\,&space;dx\:&space;\:&space;\:&space;\:&space;\:&space;v=x$ $x\text{Ei}(x)-\int&space;\frac{e^x}{x}x\,&space;dx=x\text{Ei}(x)-\int&space;e^xdx=x\text{Ei}(x)-e^x+C$

It wasn’t that hard, right? Did you use this method too? Let me know in the comments, and also tell me what you think the integrals of the other functions are!

If you liked this post give it a like, it means a lot to me! If you have any doubt or questions leave a comment and I will be happy to help! Stay tuned!

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