式展開: x√(a x^2+b x +c)の積分

x√(a x^2+b x +c)の積分

\begin{array}{rcl} \int x \sqrt{a x^{2} + b x + c} d x & = & \int x \sqrt{{(\sqrt{a} x + \frac{b}{2 \sqrt{a}})}^{2} - \frac{b^{2}}{4 a} + c} d x \dots 平 方 完 成 \\ \dots a x^{2} + b x + c = (\sqrt{a} x + α)^{2} - α^{2} + c \\ \dots (\sqrt{a} x + α)^{2} = a x^{2} + 2 \sqrt{a} α x + α^{2}, 2 \sqrt{a} α = b よ り α = \frac{b}{2 \sqrt{a}} \\ = & \int \frac{2 \sqrt{a} u - b}{2 a} \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} \frac{1}{\sqrt{a}} d u \\ \dots u = \sqrt{a} x + \frac{b}{2 \sqrt{a}} \\ \dots x = \frac{u - \frac{b}{2 \sqrt{a}}}{\sqrt{a}} = \frac{\frac{2 \sqrt{a} u - b}{2 \sqrt{a}}}{\sqrt{a}} = \frac{2 \sqrt{a} u - b}{2 a} \\ \dots \frac{d x}{d u} = \frac{1}{\sqrt{a}}, d x = \frac{1}{\sqrt{a}} d u \\ = & \frac{1}{2 a^{\frac{3}{2}}} \int (2 \sqrt{a} u - b) \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u \\ = & \frac{1}{2 a^{\frac{3}{2}}} \int 2 \sqrt{a} u \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u - \frac{1}{2 a^{\frac{3}{2}}} \int b \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u \\ = & \frac{2 \sqrt{a}}{2 a^{\frac{3}{2}}} \int u \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u - \frac{b}{2 a^{\frac{3}{2}}} \int \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u \\ = & \frac{1}{a} \int u \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u - \frac{b}{2 a^{\frac{3}{2}}} \int \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u \end{array}

\begin{array}{rcl} \frac{1}{a} \int u \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u & = & \frac{1}{a} \int u \sqrt{s} \frac{1}{2 u} d s \dots s = u^{2} - \frac{b^{2}}{4 a} + c, \frac{d s}{d u} = 2 u, d u = \frac{1}{2 u} d s \\ = & \frac{1}{2 a} \int \sqrt{s} d s \\ = & \frac{1}{2 a} [\frac{2}{3} s^{\frac{3}{2}} + C_{0}] \dots \int x^{\frac{1}{2}} d x = \frac{2}{3} x^{\frac{3}{2}} + C_{0} (C_{0} : 積 分 定 数) \\ = & \frac{1}{3 a} s^{\frac{3}{2}} + \frac{1}{2 a} C_{0} \\ = & \frac{1}{3 a} {(u^{2} - \frac{b^{2}}{4 a} + c)}^{\frac{3}{2}} + \frac{1}{2 a} C_{0} \\ = & \frac{1}{3 a} {{(\sqrt{a} x + \frac{b}{2 \sqrt{a}})}^{2} - \frac{b^{2}}{4 a} + c}^{\frac{3}{2}} + \frac{1}{2 a} C_{0} \\ = & \frac{1}{3 a} {(a x^{2} + b x + \frac{b^{2}}{4 a}) - \frac{b^{2}}{4 a} + c}^{\frac{3}{2}} + \frac{1}{2 a} C_{0} \\ = & \frac{1}{3 a} {(a x^{2} + b x + c)}^{\frac{3}{2}} + \frac{1}{2 a} C_{0} \end{array}

\begin{array}{rcl} - \frac{b}{2 a^{\frac{3}{2}}} \int \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u & = & - \frac{b}{2 a^{\frac{3}{2}}} \int \sqrt{(c - \frac{b^{2}}{4 a}) (t^{2} + 1)} \sqrt{c - \frac{b^{2}}{4 a}} d t \\ \dots t = \frac{u}{\sqrt{c - \frac{b^{2}}{4 a}}}, \frac{d t}{d u} = \frac{1}{\sqrt{c - \frac{b^{2}}{4 a}}}, d u = \sqrt{c - \frac{b^{2}}{4 a}} d t \\ \dots u^{2} + c - \frac{b^{2}}{4 a} = (u^{2} + c - \frac{b^{2}}{4 a}) \frac{c - \frac{b^{2}}{4 a}}{c - \frac{b^{2}}{4 a}} = (c - \frac{b^{2}}{4 a}) \frac{u^{2} + c - \frac{b^{2}}{4 a}}{c - \frac{b^{2}}{4 a}} = (c - \frac{b^{2}}{4 a}) (\frac{u^{2}}{c - \frac{b^{2}}{4 a}} + 1) = (c - \frac{b^{2}}{4 a}) {{(\frac{u}{\sqrt{c - \frac{b^{2}}{4 a}}})}^{2} + 1} = (c - \frac{b^{2}}{4 a}) (t^{2} + 1) \\ = & - \frac{b}{2 a^{\frac{3}{2}}} (c - \frac{b^{2}}{4 a}) \int \sqrt{t^{2} + 1} d t \\ = & - \frac{b}{2 a^{\frac{3}{2}}} (c - \frac{b^{2}}{4 a}) {\frac{1}{2} (t \sqrt{t^{2} + 1} + \ln | t + \sqrt{t^{2} + 1} | + C_{1})} \dots \int \sqrt{x^{2} + a^{2}} d x = \frac{1}{2} (x \sqrt{x^{2} + a^{2}} + a^{2} \ln | x + \sqrt{x^{2} + a^{2}} | + C_{1}) (C_{1} : 積 分 定 数) \\ = & (\frac{b^{3}}{16 a^{\frac{5}{2}}} - \frac{b c}{4 a^{\frac{3}{2}}}) (t \sqrt{t^{2} + 1} + \ln | t + \sqrt{t^{2} + 1} | + C_{1}) \\ = & (\frac{b^{3}}{16 a^{\frac{5}{2}}} - \frac{4 a b c}{16 a^{\frac{5}{2}}}) (t \sqrt{t^{2} + 1} + \ln | t + \sqrt{t^{2} + 1} | + C_{1}) \\ = & \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} (t \sqrt{t^{2} + 1} + \ln | t + \sqrt{t^{2} + 1} | + C_{1}) \\ = & \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} {(\frac{u}{\sqrt{c - \frac{b^{2}}{4 a}}}) \sqrt{{(\frac{u}{\sqrt{c - \frac{b^{2}}{4 a}}})}^{2} + 1} + \ln | (\frac{u}{\sqrt{c - \frac{b^{2}}{4 a}}}) + \sqrt{{(\frac{u}{\sqrt{c - \frac{b^{2}}{4 a}}})}^{2} + 1} | + C_{1}} \\ = & \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} {(\frac{\sqrt{a} x + \frac{b}{2 \sqrt{a}}}{\sqrt{c - \frac{b^{2}}{4 a}}}) \sqrt{{(\frac{\sqrt{a} x + \frac{b}{2 \sqrt{a}}}{\sqrt{c - \frac{b^{2}}{4 a}}})}^{2} + 1} + \ln | (\frac{\sqrt{a} x + \frac{b}{2 \sqrt{a}}}{\sqrt{c - \frac{b^{2}}{4 a}}}) + \sqrt{{(\frac{\sqrt{a} x + \frac{b}{2 \sqrt{a}}}{\sqrt{c - \frac{b^{2}}{4 a}}})}^{2} + 1} | + C_{1}} \\ = & \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} {\frac{2 a x + b}{\sqrt{4 a c - b^{2}}} \sqrt{\frac{4 a (a x^{2} + b x + c)}{4 a c - b^{2}}} + \ln | \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} + \sqrt{\frac{4 a (a x^{2} + b x + c)}{4 a c - b^{2}}} | + C_{1}} \\ \dots \frac{\sqrt{a} x + \frac{b}{2 \sqrt{a}}}{\sqrt{c - \frac{b^{2}}{4 a}}} = \frac{2 a x + b}{\sqrt{4 a c - b^{2}}} \\ \dots {(\frac{\sqrt{a} x + \frac{b}{2 \sqrt{a}}}{\sqrt{c - \frac{b^{2}}{4 a}}})}^{2} = {(\frac{2 a x + b}{\sqrt{4 a c - b^{2}}})}^{2} = \frac{4 a^{2} x^{2} + 4 a b x + b^{2}}{4 a c - b^{2}} \\ \dots {(\frac{\sqrt{a} x + \frac{b}{2 \sqrt{a}}}{\sqrt{c - \frac{b^{2}}{4 a}}})}^{2} + 1 = \frac{4 a^{2} x^{2} + 4 a b x + b^{2}}{4 a c - b^{2}} + 1 = \frac{4 a^{2} x^{2} + 4 a b x + b^{2} + 4 a c - b^{2}}{4 a c - b^{2}} = \frac{4 a^{2} x^{2} + 4 a b x + 4 a c}{4 a c - b^{2}} = \frac{4 a (a x^{2} + b x + c)}{4 a c - b^{2}} \\ = & \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} {\frac{2 a x + b}{4 a c - b^{2}} 2 \sqrt{a} \sqrt{(a x^{2} + b x + c)} + \ln | \frac{1}{\sqrt{4 a c - b^{2}}} {(2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)}} | + C_{1}} \\ = & \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} {\frac{2 a x + b}{4 a c - b^{2}} 2 \sqrt{a} \sqrt{a (a x^{2} + b x + c)} + \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | - \ln \sqrt{4 a c - b^{2}} + C_{1}} \\ = & \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \frac{2 a x + b}{4 a c - b^{2}} 2 \sqrt{a} \sqrt{(a x^{2} + b x + c)} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | - \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln \sqrt{4 a c - b^{2}} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} C_{1} \\ = & - \frac{b (2 a x + b)}{8 a^{\frac{4}{2}}} \sqrt{(a x^{2} + b x + c)} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | - \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln \sqrt{4 a c - b^{2}} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} C_{1} \\ = & - \frac{b (2 a x + b)}{8 a^{2}} \sqrt{(a x^{2} + b x + c)} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | - \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln \sqrt{4 a c - b^{2}} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} C_{1} \end{array}

\begin{array}{rcl} \int x \sqrt{a x^{2} + b x + c} d x & = & \frac{1}{a} \int u \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u - \frac{b}{2 a^{\frac{3}{2}}} \int \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u \\ = & \frac{1}{3 a} {(a x^{2} + b x + c)}^{\frac{3}{2}} + \frac{1}{2 a} C_{0} - \frac{b (2 a x + b)}{8 a^{2}} \sqrt{(a x^{2} + b x + c)} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | - \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln \sqrt{4 a c - b^{2}} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} C_{1} \\ \dots \frac{1}{a} \int u \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u = \frac{1}{3 a} {(a x^{2} + b x + c)}^{\frac{3}{2}} + \frac{1}{2 a} C_{0} \\ \dots - \frac{b}{2 a^{\frac{3}{2}}} \int \sqrt{u^{2} - \frac{b^{2}}{4 a} + c} d u = - \frac{b (2 a x + b)}{8 a^{2}} \sqrt{(a x^{2} + b x + c)} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | - \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln \sqrt{4 a c - b^{2}} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} C_{1} \\ = & \frac{1}{3 a} {(a x^{2} + b x + c)}^{\frac{3}{2}} - \frac{b (2 a x + b)}{8 a^{2}} \sqrt{(a x^{2} + b x + c)} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | + \frac{1}{2 a} C_{0} - \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln \sqrt{4 a c - b^{2}} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} C_{1} \\ = & \frac{1}{3 a} {(a x^{2} + b x + c)}^{\frac{3}{2}} - \frac{b (2 a x + b)}{8 a^{2}} \sqrt{(a x^{2} + b x + c)} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln | (2 a x + b) + 2 \sqrt{a (a x^{2} + b x + c)} | + C \dots C = \frac{1}{2 a} C_{0} - \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} \ln \sqrt{4 a c - b^{2}} + \frac{b (b^{2} - 4 a c)}{16 a^{\frac{5}{2}}} C_{1} (C : 積 分 定 数) \end{array}

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