Seeking the basic theorem of calculus

Seeking the basic theorem of calculus

If the function f (x) is continuous on [a, b] and there exists the original function f (x), then f (x) is integrable on [a, b], and B (upper limit) ∫ a (lower limit) f (x) DX = f (b) - F (a). This is Newton Leibniz formula

The basic theorem of calculus
The value of C that minimizes ∫ (10) (x ^ 2 + CX + C) ^ 2DX is ()

∫（x^2+cx+c)^2dx
=∫x^4+2cx^2(x+1)+c^2(x+1)^2 dx
=∫x^5/5+cx^4/2+2cx^3/3+c^2(x+1)^3/3+C
0 1 brought it in
The value of definite integral is 8 / 3C ^ 2 + 7 / 6C + 1 / 5
The minimum C is - 7 / 32

How is the arc length formula derived in calculus,

You are right, DS ^ 2 = DX ^ 2 + dy ^ 2ds = under the root sign (DX ^ 2 + dy ^ 2). According to this formula, you can derivate other formulas. If DX ^ 2 is put forward from the root sign, it is ∫ DS = ∫ under the root sign [1 + (dy / DX) ^ 2] * DX. Similarly, ∫ DS = ∫ under the root sign [1 + (DX / dy) ^ 2] * dy. if it is a parameter function, for t [a, b] ∫ DS