式展開: √(a x^2+b x +c)の積分

√(a x^2+b x +c)の積分

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

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