積分の公式 (1回目)
202 5 /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 sin 2 θ + cos 2 θ = 1 sin(α±β) = sinαcosβ±cosαsinβ cos(α±β) = cosαcosβ∓ sinαsinβ sin2α = 2sinαcosα cos2α = cos 2 α - sin 2 α = cos 2 α - (1 - cos 2 α) = 2cos 2 α - 1 = (1 - sin 2 α) - sin 2 α = 2sin 2 α + 1 ▼ 導出 f(x) = √(r 2 - j 2 x 2 ) 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 - sin 2 θ) = √(1 - j 2 x 2 /r 2 ) tanθ = (jx/r)/√(1 - j 2 x 2 /r 2 ) = jx/√(r 2 - j 2 x 2 ) θ = tan -1 {jx/√(r 2 - j 2 x 2 )} F 1 = ∫f(x)dx = ∫√(r 2 - j 2 x 2 )dx = ∫√(r 2 - j 2 (r/j) 2 sin 2 θ)(r/j)cosθdθ = (r/j)∫√{r 2 (1 - sin 2 θ)}cosθdθ = (r/j)∫r√(cos 2 θ)cosθdθ = (r 2 /j)∫cos 2 θdθ = (r 2 /j)(1/2)∫( cos2 θ + 1) dθ = (r 2 /j)(1/2){(1/2)sin2θ + θ} + C = (r 2 /j)(1/2){(1/2)2(sinθcosθ) + θ} + C = j(r 2 /j 2 )(1/2){sinθ√(1 - sin 2 θ) + θ} + C = j(1/2){(...
