色々な不定積分

クリスカ^{∗}

2010 年 11 月 2 日

\begin{matrix}問題 \\ a > 0 をみたす実数とするとき以下の不定積分を計算せよ. \\ (1) & ∫ \frac{1}{x^{2} + a^{2}} dx & (2) & ∫ \frac{1}{x^{2} − a^{2}} dx & (3) & ∫ \frac{1}{a^{2} − x^{2}} dx \\ (4) & ∫ \frac{1}{\sqrt{x^{2} + a^{2}}} dx & (5) & ∫ \frac{1}{\sqrt{x^{2} − a^{2}}} dx & (6) & ∫ \frac{1}{\sqrt{a^{2} − x^{2}}} dx \\ (7) & ∫ \sqrt{x^{2} + a^{2}} dx & (8) & ∫ \sqrt{x^{2} − a^{2}} dx & (9) & ∫ \sqrt{a^{2} − x^{2}} dx \end{matrix}

解答 (1) x = a tan t と置くと, dx = a sec^{2} t. よって,

∫ \frac{1}{x^{2} + a^{2}} dx = \frac{1}{a^{2}} ∫ cos^{2} t × a sec^{2} t dt

= \frac{1}{a} ∫ dt

= \frac{t}{a}

ここで, x = a tan t ⟺ t = arctan \frac{x}{a} より,

∴ ∫ \frac{1}{x^{2} + a^{2}} dx = \frac{1}{a} arctan \frac{x}{a}

∗ k-nakagawa@h6.dion.ne.jp

∫ \frac{1}{x^{2} − a^{2}} dx = ∫ \frac{1}{2a} (\frac{1}{x − a} − \frac{1}{x + a}) dx

= \frac{1}{2a} (log(x − a) − log(x + a))

= \frac{1}{2a} log |\frac{x − a}{x + a}|

(3)

∫ \frac{1}{a^{2} − x^{2}} dx = ∫ \frac{1}{2a} (\frac{1}{x + a} − \frac{1}{x − a}) dx

= \frac{1}{2a} (log(x + a) − log(x − a))

= \frac{1}{2a} log |\frac{x + a}{x − a}|

(4) x = a sinh t と置くと, dx = a cosh^{2} t. よって,

∫ \frac{1}{\sqrt{x^{2} + a^{2}}} dx = \frac{1}{a} ∫ sech ^{2}t × a cosh^{2} t dt
= ∫ dt
= t

ここで, x = a sinh t ⟺ t = arcsinh \frac{x}{a} より,

∴ ∫ \frac{1}{\sqrt{x^{2} + a^{2}}} dx = arcsinh \frac{x}{a}

x = a cosh t と置くと, dx = a sinh^{2} t. よって,

∫ \frac{1}{\sqrt{x^{2} − a^{2}}} dx = \frac{1}{a} ∫ csch ^{2}t × a sinh^{2} t dt
= ∫ dt
= t

ここで, x = a cosh t ⟺ t = arccosh \frac{x}{a} より,

∴ ∫ \frac{1}{\sqrt{x^{2} − a^{2}}} dx = arccosh \frac{x}{a}

(6) x = a sin t と置くと, dx = a cos t. よって,

∫ \frac{1}{\sqrt{a^{2} − x^{2}}} dx = \frac{1}{a} ∫ sec t × a cos t dt
= ∫ dt
= t

ここで, x = a sin t ⟺ t = arcsin \frac{x}{a} より,

∴ ∫ \frac{1}{\sqrt{a^{2} − x^{2}}} dx = arcsin \frac{x}{a}

x = a sinh t と置くと, dx = a cosh t. よって,

∫ \sqrt{x^{2} + a^{2}} dx = a ∫ cosh t × a cosh t dt

= a^{2} ∫ cosh^{2} t dt

= \frac{a^{2}}{2} ∫ (1 + cosh 2t) dt

= \frac{a^{2}}{2} (t + \frac{1}{2} sinh 2t)

= \frac{a^{2}}{2} (t + sinh t cosh t)

= \frac{a^{2}}{2} (t + sinh t \sqrt{1 + sinh^{2} t})

ここで, x = a sinh t ⟺ t = arcsinh \frac{x}{a} より,

\frac{a^{2}}{2} (t + sinh t \sqrt{1 + sinh^{2} t}) = \frac{a^{2}}{2} (arcsinh \frac{x}{a} + \frac{x}{a} \sqrt{1 + (\frac{x}{a})^{2}})

∴ ∫ \sqrt{x^{2} + a^{2}} dx = \frac{a^{2}}{2} (arcsinh \frac{x}{a} + \frac{x}{a^{2}} \sqrt{a^{2} + x^{2}})

= \frac{x}{2}\sqrt{x^{2} + a^{2}} + \frac{a^{2}}{2} arcsinh \frac{x}{a}

x = a cosh t と置くと, dx = a sinh t. よって,

∫ \sqrt{x^{2} − a^{2}} dx = a ∫ sinh t × a sinh t dt

= a^{2} ∫ sinh^{2} t dt

= \frac{a^{2}}{2} ∫ (−1 + cosh 2t) dt

= \frac{a^{2}}{2} (−t + \frac{1}{2} sinh 2t)

= \frac{a^{2}}{2} (−t + sinh t cosh t)

= \frac{a^{2}}{2} (−t + \sqrt{cosh^{2} t − 1} cosh t)

ここで, x = a cosh t ⟺ t = arccosh \frac{x}{a} より,

\frac{a^{2}}{2} (−t + \sqrt{cosh^{2} t − 1} cosh t) = \frac{a^{2}}{2} (−arccosh \frac{x}{a} + \sqrt{(\frac{x}{a} − 1)^{2}} \frac{x}{a})

∴ ∫ \sqrt{x^{2} − a^{2}} dx = \frac{a^{2}}{2} (−arccosh \frac{x}{a} + \sqrt{(\frac{x}{a} − 1)^{2}} \frac{x}{a})

= \frac{x}{2}\sqrt{x^{2} − a^{2}} − \frac{a^{2}}{2} arccosh \frac{x}{a}

x = a sin t と置くと, dx = a cos t. よって,

∫ \sqrt{a^{2} − x^{2}} dx = a ∫ cos t × a cos t dt

= a^{2} ∫ cos^{2} t dt

= \frac{a^{2}}{2} ∫ (1 + cos 2t) dt

= \frac{a^{2}}{2} (t + \frac{1}{2} sin 2t)

= \frac{a^{2}}{2} (t + sin t cos t)

= \frac{a^{2}}{2} (t + sin t \sqrt{1 − sin^{2} t})

ここで, x = a sin t ⟺ t = arcsin \frac{x}{a} より,

\frac{a^{2}}{2} (t + sin t \sqrt{1 − sin^{2} t}) = \frac{a^{2}}{2} (arcsin \frac{x}{a} + \frac{x}{a} \sqrt{1 − (\frac{x}{a})^{2}})

∴ ∫ \sqrt{a^{2} − x^{2}} dx = \frac{a^{2}}{2} (arcsin \frac{x}{a} + \frac{x}{a^{2}} \sqrt{a^{2} − x^{2}})

= \frac{x}{2}\sqrt{a^{2} − x^{2}} + \frac{a^{2}}{2} arcsin \frac{x}{a}

研究

csc θ = \frac{1}{sin θ}  sec θ = \frac{1}{cos θ}  cot θ = \frac{1}{tan θ}
sinh x = \frac{e^{x} − e^{−x}}{2}  cosh x = \frac{e^{x} + e^{−x}}{2}  tanh x = \frac{e^{x} − e^{−x}}{e^{x} + e^{−x}}

\frac{d}{dθ} sin θ = cos θ  \frac{d}{dθ} cos θ = − sin θ
\frac{d}{dx} sinh x = cosh x  \frac{d}{dx} cosh x = sinh x
sin^{2} θ + cos^{2} θ = 1  sinh^{2} x + cosh^{2} x = −1
arcsinh x = log(x + \sqrt{x^{2} + 1})  arccosh x = log(x + \sqrt{x^{2} − 1})

\frac{d}{dx} arcsin x = \frac{1}{\sqrt{1 − x^{2}}}  \frac{d}{dx} arccos x = −\frac{1}{\sqrt{1 − x^{2}}}  \frac{d}{dx} arctan x = \frac{1}{1 + x^{2}}
\frac{d}{dx}arcsinh x = \frac{1}{\sqrt{1 + x^{2}}}  \frac{d}{dx}arccosh x = \frac{1}{\sqrt{x^{2} − 1}} \frac{d}{dx}arctanh x = \frac{1}{1 − x^{2}}

研究 ; ∫ f(x) dx を計算する上で x = g(t) とおくと, dx = g′(t) dt より, ; ∫ f(x) dx = ∫ f(g(t))g′(t)dt = ∫ {F (g(t))}′ dt = F (g(t)) ; ここで F (x) は f(x) の原始関数を表す. ; 特に, g(x) が F (x) の逆関数となるとき, ; ∫ f(x) dx = ∫ dt (= t = g^{−1}(x)) ; となる.