不定積分の公式集 (deuterium)

有理関数： x²- a²型の関数	不定積分
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有理関数 • ２次式 x² ＞ a²	(x²-a²) 型	$\int {\color{Red} \frac{1}{x^2-a^2} } \, dx = \frac{1}{2a} \ln \left ( \frac{x-a}{x+a} \right ) + C = - \frac{1}{a} \tanh^{-1} \frac{x}{a} + C$ $\int {\color{Red} \frac{x}{x^2-a^2} } \, dx = \frac{1}{2} \ln (x^2-a^2) + C$ $\int {\color{Red} \frac{x^2}{x^2-a^2} } \, dx = x + \frac{a}{2} \ln \left ( \frac{x-a}{x+a} \right ) + C$ $\int {\color{Red} \frac{x^3}{x^2-a^2} } \, dx = \frac{x^2}{2} + \frac{a^2}{2} \ln (x^2-a^2) + C$ $\int {\color{Red} \frac{1}{x(x^2-a^2)} } \, dx = \frac{1}{2a^2} \ln \left ( \frac{x^2-a^2}{x^2} \right ) + C$ $\int {\color{Red} \frac{1}{x^2(x^2-a^2)} } \, dx = \frac{1}{a^2x} + \frac{1}{2a^3} \ln \left ( \frac{x-a}{x+a} \right ) + C$ $\int {\color{Red} \frac{1}{x^3(x^2-a^2)} } \, dx = \frac{1}{2a^2x^2} + \frac{1}{2a^4} \ln \left ( \frac{x^2-a^2}{x^2} \right ) + C$
	(x²-a²)²型	$\int {\color{Red} \frac{1}{(x^2-a^2)^2} } \, dx = - \frac{x}{2a^2(x^2-a^2)} - \frac{1}{4a^3} \ln \left ( \frac{x-a}{x+a} \right ) + C$ $\int {\color{Red} \frac{x}{(x^2-a^2)^2} } \, dx = - \frac{1}{2(x^2-a^2)} + C$ $\int {\color{Red} \frac{x^2}{(x^2-a^2)^2} } \, dx = - \frac{x}{2(x^2-a^2)} + \frac{1}{4a} \ln \left ( \frac{x-a}{x+a} \right ) + C$ $\int {\color{Red} \frac{x^3}{(x^2-a^2)^2} } \, dx = - \frac{a^2}{2(x^2-a^2)} + \frac{1}{2} \ln (x^2-a^2) + C$ $\int {\color{Red} \frac{1}{x(x^2-a^2)^2} } \, dx = - \frac{1}{2a^2(x^2-a^2)} - \frac{1}{2a^4} \ln \left ( \frac{x^2-a^2}{x^2} \right ) + C$ $\int {\color{Red} \frac{1}{x^2(x^2-a^2)^2} } \, dx = - \frac{1}{a^4x} - \frac{x}{2a^4(x^2-a^2)} - \frac{3}{4a^5} \ln \left ( \frac{x-a}{x+a} \right ) + C$ $\int {\color{Red} \frac{1}{x^3(x^2-a^2)^2} } \, dx = - \frac{1}{2a^4x^2} - \frac{1}{2a^4(x^2-a^2)} - \frac{1}{a^6} \ln \left ( \frac{x^2-a^2}{x^2} \right ) + C$
	(x²-a²)ⁿ型	$\int {\color{Red} \frac{1}{(x^2-a^2)^n} } \, dx = - \frac{x}{2(n-1)a^2(x^2-a^2)^{n-1}} - \frac{2n-3}{(2n-2)a^2} \int \frac{1}{(x^2-a^2)^{n-1}} \, dx$ $\int {\color{Red} \frac{x}{(x^2-a^2)^n} } \, dx = - \frac{1}{2(n-1)} \frac{1}{(x^2-a^2)^{n-1}} + C$ $\int {\color{Red} \frac{1}{x(x^2-a^2)^n} } \, dx = - \frac{1}{2(n-1)a^2(x^2-a^2)^{n-1}} - \frac{1}{a^2} \int \frac{1}{x(x^2-a^2)^{n-1}} \, dx$ $\int {\color{Red} \frac{x^m}{(x^2-a^2)^n} } \, dx =\int \frac{x^{m-2}}{(x^2-a^2)^{n-1}} \, dx + a^2 \int \frac{x^{m-2}}{(x^2-a^2)^n} \, dx$ $\int {\color{Red} \frac{1}{x^m(x^2-a^2)^n} } \, dx = \frac{1}{a^2} \int \frac{1}{x^{m-2}(x^2-a^2)^n} \, dx - \frac{1}{a^2} \int \frac{1}{x^m(x^2-a^2)^{n-1}} \, dx$

有理関数 • ２次式 x² ＜ a²	(a²-x²) 型	$\int {\color{Red} \frac{1}{a^2-x^2} } \, dx = - \frac{1}{2a} \ln \left ( \frac{a-x}{a+x} \right ) + C =\frac{1}{a} \tanh^{-1} \frac{x}{a} + C$ $\int {\color{Red} \frac{x}{a^2-x^2} } \, dx = - \frac{1}{2} \ln (a^2-x^2) + C$ $\int {\color{Red} \frac{x^2}{a^2-x^2} } \, dx = - x - \frac{a}{2} \ln \left ( \frac{a-x}{a+x} \right ) + C$ $\int {\color{Red} \frac{x^3}{a^2-x^2} } \, dx = - \frac{x^2}{2} - \frac{a^2}{2} \ln (a^2-x^2) + C$ $\int {\color{Red} \frac{1}{x(a^2-x^2)} } \, dx = - \frac{1}{2a^2} \ln \left ( \frac{a^2-x^2}{x^2} \right ) + C$ $\int {\color{Red} \frac{1}{x^2(a^2-x^2)} } \, dx = - \frac{1}{a^2x} - \frac{1}{2a^3} \ln \left ( \frac{a-x}{a+x} \right ) + C$ $\int {\color{Red} \frac{1}{x^3(a^2-x^2)} } \, dx = - \frac{1}{2a^2x^2} - \frac{1}{2a^4} \ln \left ( \frac{a^2-x^2}{x^2} \right ) + C$
	(a²-x²)²型	$\int {\color{Red} \frac{1}{(a^2-x^2)^2} } \, dx = \frac{x}{2a^2(a^2-x^2)} - \frac{1}{4a^3} \ln \left ( \frac{a-x}{a+x} \right ) + C$ $\int {\color{Red} \frac{x}{(a^2-x^2)^2} } \, dx = \frac{1}{2(a^2-x^2)} + C$ $\int {\color{Red} \frac{x^2}{(a^2-x^2)^2} } \, dx = \frac{x}{2(a^2-x^2)} + \frac{1}{4a} \ln \left ( \frac{a-x}{a+x} \right ) + C$ $\int {\color{Red} \frac{x^3}{(a^2-x^2)^2} } \, dx = \frac{a^2}{2(a^2-x^2)} + \frac{1}{2} \ln(a^2-x^2) + C$ $\int {\color{Red} \frac{1}{x(a^2-x^2)^2} } \, dx = \frac{1}{2a^2(a^2-x^2)} - \frac{1}{2a^4} \ln \left ( \frac{a^2-x^2}{x^2} \right ) + C$ $\int {\color{Red} \frac{1}{x^2(a^2-x^2)^2} } \, dx = - \frac{1}{a^4x} + \frac{x}{2a^4(a^2-x^2)} + \frac{3}{4a^5} \ln \left ( \frac{a+x}{a-x} \right ) + C$ $\int {\color{Red} \frac{1}{x^3(a^2-x^2)^2} } \, dx = - \frac{1}{2a^4x^2} + \frac{1}{2a^4(a^2-x^2)} - \frac{1}{a^6} \ln \left ( \frac{a^2-x^2}{x^2} \right ) + C$
	(a²-x²)ⁿ型	$\int {\color{Red} \frac{1}{(a^2-x^2)^n} } \, dx = \frac{x}{2(n-1)a^2(a^2-x^2)^{n-1}} + \frac{2n-3}{(2n-2)a^2} \int \frac{1}{(a^2-x^2)^{n-1}} \, dx$ $\int {\color{Red} \frac{x}{(a^2-x^2)^n} } \, dx = \frac{1}{2(n-1)(a^2-x^2)^{n-1}} + C$ $\int {\color{Red} \frac{1}{x(a^2-x^2)^n} } \, dx = \frac{1}{2(n-1)a^2(a^2-x^2)^{n-1}} + \frac{1}{a^2} \int \frac{1}{x(a^2-x^2)^{ n-1}} \, dx$ $\int {\color{Red} \frac{x^m}{(a^2-x^2)^n} } \, dx = a^2 \int \frac{x^{m-2}}{(a^2-x^2)^n} \, dx - \int \frac{x^{m-2}}{(a^2-x^2)^{n-1}} \, dx$ $\int {\color{Red} \frac{1}{x^m(a^2-x^2)^n} } \, dx = \frac{1}{a^2} \int \frac{1}{x^m(a^2-x^2)^{n-1}} \, dx + \frac{1}{a^2} \int \frac{1}{x^{m-2}(a^2-x^2)^n} \, dx$