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α + t 2 dt = t α + t 2 −
α 1
+ t2 dt =
2
α = 1
lunghezza(γ) = 2
1
0
α + t 2 dt + α log |t + α + t 2 | + c,
t α + t2 + α log |t + α + t2
| + c .
1
0
1 + t 2 dt = √ 2 + log(1 + √ 2).
s = t2 t = √ s dt = ds
2 √ s
t
I
2 1 + t2 dt = 1
1
1 1 1
s(1 + s) ds = (s2 + s + 1/4) − 1/4 ds =
2 0
2 0
2
= 1
1
3 1
(2s + 1) 2 − 1 ds = w2 − 1 dw,
4
8
0
1
1
0
(s + 1/2) 2 − 1/4 ds
w = 2s + 1 = 2t2 + 1 α = −1
t
I
2 1 + t2 dt = 1
3
1 1
w2 − 1 dw = w
8 1
8 2
w2 − 1 − log(w + w2
− 1)
w=3 w=1
= 1
16 (3√8 − log(3 + √ 8)) = 1
3
8
√ 2 − 1
2 log(1 + √ 2) 2
= 1
3
8
√ 2 − log(1 + √
2) ,
(1 + √ 2) 2 = 3 + √ 8 s = t2
2t
I
1 + t2 1
dt = (1 + s)
0
1/2 2
ds = v
1
1/2 dv = 2
3 [v3/2 ] 2 v=1 = 2
3 (2√2 − 1).
G :=
=
=
1
lunghezza(γ)
2
√ 2 + log(1 + √ 2)
1
√ 2 + log(1 + √ 2)
x(t) ˙x 2 (t) + ˙y 2
(t) dt,
I
1
0
1
4
t 2 1 + t 2 dt,
1
3 √ 2 − log(1 + √ 2)
0
I
y(t) ˙x 2 (t) + ˙y 2
(t) dt
2t 1 + t2
dt
, 4
3 (2√
2 − 1)
γ ρ(θ) = 3θ θ ∈ I := [0, 5π]
γ
µ : R 2 → [0, +∞[
µ(x, y) = 6(x 2 + y 2 ) γ
x(θ) = ρ(θ) cos θ = 3θ cos θ,
y(θ) = ρ(θ) sin θ = 3θ sin θ,
˙x 2 (θ) + ˙y 2 (θ) = ( ˙ρ(θ) cos θ − ρ(θ) sin θ) 2 + ( ˙ρ(θ) sin θ + ρ(θ) cos θ) 2
= ˙ρ 2 (θ) cos2 θ + ρ2 (θ) sin2 θ − 2 ˙ρ(θ)ρ(θ) sin θ cos θ + ˙ρ 2 (θ) sin2 θ + ρ2 (θ) cos2 θ + 2 ˙ρ(θ)ρ(θ) sin θ cos θ
= ˙ρ 2 (θ) + ρ 2 (θ).
˙x 2 (θ) + ˙y 2 (θ) = 3 1 + θ 2 .