置換積分のコツ

高校レベルの置換積分の代表的な例を集めました。

Transcript

1.   ∫ f(cos θ, sin θ) dθ ͷ࣌ʹ͸ɼt = tan θ 2 ͷஔ׵Λ͢Δɻ   tan θ = sin(θ 2 + θ 2 ) cos(θ 2 + θ 2 ) = 2 sin θ 2 cos θ 2 cos2 θ 2 − sin2 θ 2 = 2 tan θ 2 1 − tan2 θ 2 ʢ෼฼ͱ෼ࢠΛ cos2 θ 2 Ͱׂͬͨʣ = 2t 1 − t2 ͜͜Ͱɼਤܗͱͯ͠ͷ tan θ ͷఆٛʹΑͬͯɼҰͭͷ͕֯ θɼఈล 1 − t2ɼߴ͞ 2t ͷ௚֯ࡾ֯ܗΛඳ͘ɻࡾฏํͷఆཧΑΓࣼล͸        θ 1 − t2 2t l l = √ (1 − t2)2 + (2t)2 = √ (1 + 2t2 + t4) = (1 + t2) ਤΑΓɼ cos θ = 1 − t2 1 + t2 sin θ = 2t 1 + t2 sin θ ͷ྆ลΛ θ Ͱඍ෼ͯ͠ɼ cos θ = dt dθ · d dt ( 2t 1 + t2 ) = dt dθ · 2 · (1 + t2) − 2t · 2t (1 + t2)2 1 − t2 1 + t2 = dt dθ · 2(1 − t2) (1 + t2)2 dθ = 2dt 1 + t2 ·ͨɼcos θ ͷࣜʹண໨͠ɼ (1 + t2) cos θ = 1 − t2 (1 + cos θ)t2 = 1 − cos θ t = √ 1 − cos θ 1 + cos θ
2.   ∫ f(x2)x dx ͷ࣌ʹ͸ɼt = x2 ͷஔ׵Λ͢Δɻ   dt = 2xdx ∫ f(x2)dx = ∫ f(t) dt 2 = 1 2 F(t) + C = 1 2 F(x2) + C   ∫ f(sin θ) cos θ dθ ͷ࣌ʹ͸ɼt = sin θ ͷஔ׵Λ͢Δɻ   dt = cos θdθ ∫ f(sin θ) cos θdθ = ∫ f(t)dt = F(t) + C = F(sin θ) + C   ∫ f(cos θ) sin θ dθ ͷ࣌ʹ͸ɼt = cos θ ͷஔ׵Λ͢Δɻ   dt = − sin θdθ ∫ f(cos θ) sin θdθ = − ∫ f(t)dt = −F(t) + C = −F(cos θ) + C
3.   ∫ f(x)f (x) dx ͷ࣌ʹ͸ɼt = f(x) ͷஔ׵Λ͢Δɻ   dt = f (x) dx ∫ f(x)f (x) dx = ∫ t dt = 1 2 t2 + C = 1 2 (f(x))2 + C   ∫ f (x) f(x) dx ͷ࣌ʹ͸ɼt = f(x) ͷஔ׵Λ͢Δɻ   dt = f (x) dx ∫ f (x) f(x) dx = ∫ 1 t dt = log |t| + C = log |f(x)| + C   ∫ f (x) √ f(x) dx ͷ࣌ʹ͸ɼt = √ f(x) ͷஔ׵Λ͢Δɻ   dt dx = 1 2 (f(x))−1 2 f (x) 2dt = f (x) √ f(x) dx ∫ f (x) √ f(x) dx = 2 ∫ dt = 2t + C = 2 √ f(x) + C