Integral of Si(x) and Ci(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 functions Si(x) and Ci(x). If you gave it a try, let me know in the comments if you used this approach!

Integral of Si(x)

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

U-substitution won’t work, so let’s use integration by parts. We know that Si(x) is given by the integral of sin(x)/x, therefore the derivative of Si(x) is sin(x)/x.

$u&space;=&space;\text{Si}(x)\;&space;\;&space;\;&space;\;&space;\;&space;dv=dx$ $du&space;=\frac{\sin(x)}{x}dx\;&space;\;&space;\;&space;\;&space;\;&space;v=x$ $x\text{Si}(x)-\int&space;\frac{\sin(x)}{x}x\,&space;dx=x\text{Si}(x)-\int\sin(x)dx=x\text{Si}(x)+\cos(x)+C$

Integral of Ci(x)

For this we are going to use the same procedure.

$\int&space;\text{Si}(x)\,&space;dx$ $u=\text{Si}(x)\;&space;\;&space;\;&space;\;&space;\;&space;dv=dx$ $du=\frac{\cos(x)}{x}dx\;&space;\;&space;\;&space;\;&space;\;&space;v=x$ $x\text{Ci}(x)-\int&space;\frac{\cos(x)}{x}x\,&space;dx=x\text{Ci}(x)-\int\cos(x)dx=x\text{Ci}(x)-\sin(x)+C$

What do you think the integrals of S(x) and C(x) are? Give it a try and write the answer in the comments!

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