積分の公式 (1回目)

2025/10/15(水)
積分の公式 (1回目)
 
(integral)
 
■ 積分(公式)
▼ 三角関数の公式
(sinθ)' =  cosθ
(cosθ)' = -sinθ
sin²θ + cos²θ = 1
sin(α±β) = sinαcosβ±cosαsinβ
cos(α±β) = cosαcosβ∓ sinαsinβ
sin2α = 2sinαcosα
cos2α = cos²α - sin²α
= cos²α - (1 - cos²α) = 2cos²α - 1
= (1 - sin²α) - sin²α = 2sin²α + 1
 
▼ 導出
f(x) = √(r² - j²x²)
 
x =(r/j)sinθと置く
dx/dθ = (r/j)cosθ, dx = (r/j)cosθdθ
 
θ = sin^-1(jx/r)
sinθ = jx/r, cosθ = √(1 - sin²θ) = √(1 - j²x²/r²)
tanθ = (jx/r)/√(1 - j²x²/r²) = jx/√(r² - j²x²)
θ = tan^-1{jx/√(r² - j²x²)}
 
F₁ = ∫f(x)dx = ∫√(r² - j²x²)dx
= ∫√(r² - j²(r/j)²sin²θ)(r/j)cosθdθ
= (r/j)∫√{r²(1 - sin²θ)}cosθdθ
= (r/j)∫r√(cos²θ)cosθdθ
= (r²/j)∫cos²θdθ
= (r²/j)(1/2)∫(cos2θ + 1)dθ
= (r²/j)(1/2){(1/2)sin2θ + θ} + C
= (r²/j)(1/2){(1/2)2(sinθcosθ) + θ} + C
= j(r²/j²)(1/2){sinθ√(1 - sin²θ) + θ} + C
= j(1/2){(r/j)sinθ√((r²/j²) - (r²/j²)sin²θ) + (r²/j²)θ} + C
= j(1/2){x√((r²/j²) - x²) + (r²/j²)sin^-1(jx/r)} + C
= (1/2){x√(r² - j²x²) + (r²/j)sin^-1(jx/r)} + C
= (1/2){x√(r² - j²x²) + (r²/j)tan^-1{jx/√(r² - j²x²)} + C
 
F₂ = ∫1/f(x)dx = ∫{1/√(r² - j²x²)}dx
= ∫{1/√(r² - j²(r/j)²sin²θ)}(r/j)cosθdθ
= (r/j)∫{1/√{r²(1 - sin²θ)}cosθdθ
= (1/j)∫{1/√(cos²θ)}cosθdθ
= (1/j)∫dθ
= θ/j + C
= (1/j)sin^-1(jx/r) + C
= (1/j)tan^-1{jx/√(r² - j²x²)} + C
 
 
■ 結果
▼ 定義
f(x) = √(r² - j²x²)
F₁ = ∫f(x)dx = ∫√(r² - j²x²)dx
F₂ = ∫1/f(x)dx = ∫{1/√(r² - j²x²)}dx
sin^-1(jx/r) = tan^-1{jx/√(r² - j²x²)}
 
▼ 公式
∫√(r² - j²x²)dx = (1/2){x√(r² - j²x²) + (r²/j)sin^-1(jx/r)} + C
∫{1/√(r² - j²x²)}dx = (1/j)sin^-1(jx/r) + C