The second fundamental theorem of Calculus.Larson Hostetler Edwards Calculus sixth ed pg. 280
b
∫
a
x
∫
a
There is a circle of understanding / madness in the “picture” above of two equations. They are text and not active code. Before computers the same struggle to choose the letters and symbolism were also an issue! Even though they where written on paper with feathers and ink, sent by horseback and sail ship to mathematicians around the world, the problem was how to write the formulas in a conventional way that also kept the rigorous logic and computations intact. Consider the capitalization (upper case) of the F . Like a proper noun, capitol (upper case) F was understood as function but also a name, or more strictly a title in an hierarchy of sentence structure.
v[t]:=3t^2-12t+9
Integrate[v[t],t]
9t-6+
2
t
3
t
When companies like Texas Instruments and Casio introduced “pretty print” many could not believe the little equations on the tiny screens actually appeared like the text in our books. Now they do so in flying colors that actively calculate, will not evaluate, or, be assigned rules and user names.
∫v[t]t
9t-6+
2
t
3
t
Position==9t-6+
2
t
3
t
Position9t-6+
2
t
3
t
Sum[f[x,y],{x,3},{y,3}]
f[1,1]+f[1,2]+f[1,3]+f[2,1]+f[2,2]+f[2,3]+f[3,1]+f[3,2]+f[3,3]
HoldFormLimit,x1
x^2-1
x-1
x1
2
x
x-1
TraditionalForm[(y+3)^2/((y-2)(y+5))]
2
(y+3)
(y-2)(y+5)
There are hundreds of lines of code supporting the “integrate function” https://www.wolfram.com/wolfram-u/catalog/edu022/
Consider the definition from the Mathematica resource center of NestList.
NestList[f, expr, n] gives a list of the results of applying f to expr 0 through n times
Clear[f]NestList[f,x,5]
{x,f[x],f[f[x]],f[f[f[x]]],f[f[f[f[x]]]],f[f[f[f[f[x]]]]]}