Antiderivação e o problema da constante

𝑓𝑥 𝐼

ℝ

𝑓𝑥 𝐼

𝑓𝑥 𝔗𝐼 (𝑓𝑥 )

𝑔𝑥

𝐷 ʿ𝑔𝑥 𝑔𝑥 , ..

𝑓𝑥 ,

𝑥 𝑓𝑥 𝑓

{(𝑎1 ; 𝑎12 ), . . , (𝑎𝑛 ; 𝑎𝑛2 ), . . }

𝑓= 𝑓𝑥 = 𝑥 2

λ𝑥 (𝑥 2 ) 𝑦𝑥 𝐼

𝑔𝑥

𝐷 ʿ𝑦𝑥 = 𝐷 ʿ𝑔𝑥

𝐶

𝑦𝑥 = 𝑔𝑥 + 𝐶 ℎ𝑥 𝑦𝑥 − 𝑔𝑥

𝐼

ℎ𝑥 =

𝐷 ʿℎ𝑥 = 𝐷 ʿ𝑦𝑥 − 𝐷 ʿ𝑔𝑥

𝐷 ʿ𝑦𝑥 = 𝐷 ʿ𝑔𝑥

𝐷 ʿℎ𝑥 = 0

ℎ𝑥 = 𝐶 𝐶 = 𝑦𝑥 − 𝑔𝑥

𝐶 + 𝑔𝑥 = 𝑦𝑥

ℎ𝑥 𝐷 ʿ𝑦𝑥 = 𝐷 ʿ𝑔𝑥

ℎ𝑥

𝐹𝑥 𝐼

𝑓𝑥 𝑓𝑥

𝐹𝑥 + 𝐶 𝑓𝑥

𝐶

𝐼

𝐹𝑥 + 𝐶

𝑔𝑥 𝑓𝑥

𝐼 𝑓𝑥

𝐷 ʿ𝑔𝑥 = 𝑓𝑥 𝐼

𝐹𝑥

𝐷 ʿ𝐹𝑥 = 𝑓𝑥 = 𝐷 ʿ𝑔𝑥 𝑔𝑥 = 𝐹𝑥 + 𝐶 𝐶

𝐶

1

2 𝐹𝑥 𝐼

wx ϵ 𝔗I (fx )

D ʿwx = fx

w x = Fx + C D ʿwx = D ʿFx + C = D ʿFx + D ʿC = D ʿFx = fx

Fx ϵ 𝔗I (fx )

Fx

fx wx = Fx + C = (Fx + C) + C

Fx = Fx + C

wx = (Fx + C) + C

Fx

wx = {((Fx + C) + C) + C}

wx ϵ 𝔗I (fx )

wx = lim (Fx + λC) = Fx + lim λC λ→∞

λ→∞

lim λC

λ→∞

wx ϵ 𝔗I (fx ) lim λC

λ→∞

wx = lim (Fx + λC) = Fx + λ→∞

lim λC = δ < ∞

λ→∞

w x = Fx + δ D ʿwx = D ʿFx + δ = D ʿFx + D ʿδ = D ʿFx = fx wx ϵ 𝔗I (fx ) lim λC

λ→∞

lim λC = ∞

λ→∞

wx = lim (Fx + λC) = Fx + λ→∞

w x = Fx + ∞ = ∞

wx

wx 𝔗I (fx ) = Λ hx 𝔗I (fx )

𝐹𝑥 𝐼

𝔗𝐼 (𝑓𝑥 ) 𝑓𝑥

𝔗𝐼 (𝑓𝑥 )

𝐹𝑥

ℎ𝑥 = 𝑤𝑥 − 𝑔𝑥

𝔗𝐼 (𝑓𝑥 ) 𝔗𝐼 (𝑓𝑥 ) 𝔗𝐼 (𝑓𝑥 )