# Special functions (integrals)

U-substitution, integration by parts and trigonometric substitution are the most used techniques for integration. When nothing is going to work in the case of a certain function, we’ll say that that function doesn’t have an elementary antiderivative, which means there is no function whose derivative gives the function we want to integrate. This is why special functions have been invented.

$\int&space;\frac{e^x}{x}dx$

Although this looks innocent, you’ve got to believe me when I say that no matter what you do, you will never solve this. Since the antiderivative is a function we don’t know (like log, ln, cos, sin, etc…) we call it Ei (exponential integral function).

$\int&space;\frac{e^x}{x}dx=\text{Ei}(x)+C$

Therefore

$\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}\text{Ei}(x)=\frac{e^x}{x}$

The other special integrals are:

Logarithmic integral function

$\int\frac{1}{\ln(x)}dx=\text{li}(x)+C$

Eulerian logarithmic integral

$\text{Li}(x)=\text{li}(x)-\text{li}(2)$

Error function

$\int&space;e^{-x^2}dx=\text{erf}(x)+C$

Sine integral function (1)

$\int\frac{\sin(x)}{x}dx=\text{Si}(x)+C$

Cosine integral function (1)

$\int\frac{\cos(x)}{x}dx=\text{Ci}(x)+C$

Sine integral function (2)

$\int&space;\sin(x^2)dx=\text{S}(x)+C$

Cosine integral function (2)

$\int&space;\cos(x^2)dx=\text{C}(x)+C$

So, if you see these integrals, you won’t have to try the impossible for nothing, you just have to remember the name antiderivative. In the next post we’ll see how to integrate these special functions! Sounds fun, doesn’t it? But I would like you to give ’em a try first!

$\int&space;\text{Ei}(x)dx$ $\int&space;\text{li}(x)dx$ $\int&space;\text{Li}(x)dx$ $\int&space;\text{erf}(x)dx$ $\int&space;\text{Si}(x)dx$ $\int&space;\text{Ci}(x)dx$ $\int&space;\text{S}(x)dx$ $\int&space;\text{C}(x)dx$

Don’t worry if you don’t know how to solve them; i will show you in another post!

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