積分の公式 (2回目)

2025/10/20(月)
積分の公式 (2回目)
 
(integral)
 
■ 積分(応用)
▼ 三角関数
f(θ) = cosθ + isinθ
f'(θ) = -sinθ + icosθ
if'(θ) = -isinθ - cosθ = -f(θ)
f(θ) = -if'(θ)  … ①
f(θ) = exp(iθ)と置くと
f'(θ) = iexp(iθ)
if'(θ) = -exp(iθ) = -f(θ)
f(θ) = -if'(θ)
で式①を満たす
 
exp(iθ)  = cosθ + isinθ
exp(-iθ) = cosθ - isinθ
sinθ = {exp(iθ) - exp(-iθ)}/(2i)
cosθ = {exp(iθ) + exp(-iθ)}/2
tanθ = {exp(iθ) - exp(-iθ)}/{exp(iθ) + exp(-iθ)}i
 
▼ 双曲線関数
sinhθ = {exp(θ) - exp(-θ)}/2
coshθ = {exp(θ) + exp(-θ)}/2
tanhθ = {exp(θ) - exp(-θ)}/{exp(θ) + exp(-θ)}
 
▼ 混合
isin(iθ) = {exp(-θ)-exp(θ)}/2 = -sinhθ
 cos(iθ) = {exp(-θ)+exp(θ)}/2 =  coshθ
itan(iθ) = {exp(-θ)-exp(θ)}/{exp(-θ)+exp(θ)} = -tanhθ
 
▼ 双曲線関数
y = tanhθ = -itan(iθ)
iy = tan(iθ)
iθ = tan^-1(iy)
θ = -itan^-1(iy) = tanh^-1y
 
▼ √(r² + x²)
∫√(r² + x²)dx
∫√(r² - j²x²)dx = (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
 
j = i, -itan^-1(iy) = tanh^-1y
 
∫√(r² - j²x²)dx = ∫√(r² + x²)dx
= (1/2){x√(r² - i²x²) + (r²/i)tan^-1{ix/√(r² - i²x²)}} + C
= (1/2){x√(r² + x²) - r²itan^-1{ix/√(r² + x²)}} + C
= (1/2){x√(r² + x²) + r²tanh^-1{x/√(r² + x²)}} + C
 
▼ √(r² - x²)
∫√(r² - x²)dx
∫√(r² - j²x²)dx = (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
 
j = 1
 
∫√(r² - j²x²)dx = ∫√(r² - x²)dx
= (1/2){x√(r² - 1²x²) + (r²/1)tan^-1{1･x/√(r² - 1²x²)}} + C
= (1/2){x√(r² - x²) + r²tan^-1{x/√(r² - x²)}} + C
 
▼ 1/√(r² + x²)
∫{1/√(r² + x²)}dx
∫{1/√(r² - j²x²)}dx = (1/j)sin^-1(jx/r) + C
= (1/j)tan^-1{jx/√(r² - j²x²)} + C
 
j = i, -itan^-1(iy) = tanh^-1y
 
∫{1/√(r² - j²x²)}dx = ∫{1/√(r² + x²)}dx
= (1/i)tan^-1{ix/√(r² - i²x²)} + C
= -itan^-1{ix/√(r² + x²)} + C
= tanh^-1{x/√(r² + x²)} + C
 
▼ 1/√(r² - x²)
∫{1/√(r² - x²)dx}
∫{1/√(r² - j²x²)}dx = (1/j)sin^-1(jx/r) + C
= (1/j)tan^-1{jx/√(r² - j²x²)} + C
 
j = 1
 
∫{1/√(r² - j²x²)}dx = ∫{1/√(r² - x²)}dx
= (1/1)tan^-1{1･x/√(r² - 1²x²)} + C
= tan^-1{x/√(r² - x²)} + C
 
 
■ 結果
▼ 定義域
xの定義域に注意
 
▼ 公式
∫√(r² + x²)dx = (1/2){x√(r² + x²) + r²tanh^-1(x/√(r² + x²))} + C
∫√(r² - x²)dx = (1/2){x√(r² - x²) + r²tan^-1(x/√(r² - x²))} + C
∫{1/√(r² + x²)}dx = tanh^-1{x/√(r² + x²)} + C
∫{1/√(r² - x²)}dx = tan^-1{x/√(r² - x²)} + C