式展開: √(x^2+a^2)の積分

√(x^2+a^2)の積分

\begin{array}{rcl} \int \sqrt{x^{2} + a^{2}} d x & = & \int \sqrt{a^{2} \tan^{2} (θ) + a^{2}} a \frac{1}{\cos^{2} (θ)} d θ \dots x = a \tan (θ), \frac{d x}{d θ} = a \frac{d}{d θ} \tan (θ) = a \frac{1}{\cos^{2} (θ)}, d x = a \frac{1}{\cos^{2} (θ)} d θ \\ = & a \int \sqrt{a^{2} {\tan^{2} (θ) + 1}} \frac{1}{\cos^{2} (θ)} d θ \\ = & a \int a \sqrt{\tan^{2} (θ) + 1} \frac{1}{\cos^{2} (θ)} d θ \\ = & a^{2} \int \sqrt{\frac{1}{\cos^{2} (θ)}} \frac{1}{\cos^{2} (θ)} d θ \dots \tan^{2} (θ) + 1 = \frac{\sin^{2} (θ)}{\cos^{2} (θ)} + 1 = \frac{\sin^{2} (θ) + \cos^{2} (θ)}{\cos^{2} (θ)} = \frac{1}{\cos^{2} (θ)} \\ = & a^{2} \int \frac{1}{\cos (θ)} \frac{1}{\cos^{2} (θ)} d θ \\ = & a^{2} \int \frac{1}{\cos^{3} (θ)} d θ \\ = & a^{2} [\frac{1}{2} {\tan (θ) \frac{1}{\cos (θ)} + \ln | \tan (θ) + \frac{1}{\cos (θ)} |} + C_{0}] \dots \int \frac{1}{\cos^{3} (θ)} d θ = \frac{1}{2} {\tan (θ) \frac{1}{\cos (θ)} + \ln | \tan (θ) + \frac{1}{\cos (θ)} |} + C_{0} (C_{0} : 積 分 定 数) \\ = & \frac{a^{2}}{2} {\tan (θ) \frac{1}{\cos (θ)} + \ln | \tan (θ) + \frac{1}{\cos (θ)} | + C_{0}} \\ = & \frac{1}{2} {a^{2} \tan (θ) \frac{1}{\cos (θ)} + a^{2} \ln | \tan (θ) + \frac{1}{\cos (θ)} | + a^{2} C_{0}} \\ = & \frac{1}{2} {a^{2} \frac{1}{a^{2}} x \sqrt{x^{2} + a^{2}} + a^{2} \ln | \frac{1}{a} (x + \sqrt{x^{2} + a^{2}}) | + a^{2} C_{0}} \dots \tan (θ) \frac{1}{\cos (θ)} = \frac{1}{a^{2}} x \sqrt{x^{2} + a^{2}}, \tan (θ) + \frac{1}{\cos (θ)} = \frac{1}{a} (x + \sqrt{x^{2} + a^{2}}) \\ = & \frac{1}{2} [x \sqrt{x^{2} + a^{2}} + a^{2} {\ln | x + \sqrt{x^{2} + a^{2}} | - \ln | a |} + a^{2} C_{0}] \dots \ln \frac{A}{B} = \ln A - \ln B \\ = & \frac{1}{2} {x \sqrt{x^{2} + a^{2}} + a^{2} \ln | x + \sqrt{x^{2} + a^{2}} | - a^{2} \ln | a | + a^{2} C_{0}} \\ = & \frac{1}{2} {x \sqrt{x^{2} + a^{2}} + a^{2} \ln | x + \sqrt{x^{2} + a^{2}} |} - \frac{a^{2}}{2} \ln | a | + \frac{a^{2}}{2} C_{0} \\ = & \frac{1}{2} {x \sqrt{x^{2} + a^{2}} + a^{2} \ln | x + \sqrt{x^{2} + a^{2}} |} + C \dots C = - \frac{a^{2}}{2} \ln | a | + \frac{a^{2}}{2} C_{0} (C : 積 分 定 数) \end{array}

\begin{array}{rcl} \tan (θ) \frac{1}{\cos (θ)} & = & \tan (θ) \sqrt{\frac{1}{\cos^{2} (θ)}} \\ = & \tan (θ) \sqrt{\tan^{2} (θ) + 1} \dots \frac{1}{\cos^{2} (θ)} = \frac{\sin^{2} (θ) + \cos^{2} (θ)}{\cos^{2} (θ)} = \frac{\sin^{2} (θ)}{\cos^{2} (θ)} + 1 = \tan^{2} (θ) + 1 \\ = & \frac{x}{a} \sqrt{{(\frac{x}{a})}^{2} + 1} \dots x = a \tan (θ) よ り \tan (θ) = \frac{x}{a} \\ = & \frac{x}{a} \sqrt{\frac{x^{2} + a^{2}}{a^{2}}} \\ = & \frac{x}{a} \sqrt{\frac{1}{a^{2}} (x^{2} + a^{2})} \\ = & \frac{x}{a} \frac{1}{a} \sqrt{x^{2} + a^{2}} \\ = & \frac{1}{a^{2}} x \sqrt{x^{2} + a^{2}} \end{array}

\begin{array}{rcl} \tan (θ) + \frac{1}{\cos (θ)} & = & \tan (θ) + \sqrt{\frac{1}{\cos^{2} (θ)}} \\ = & \tan (θ) + \sqrt{\tan^{2} (θ) + 1} \dots \frac{1}{\cos^{2} (θ)} = \frac{\sin^{2} (θ) + \cos^{2} (θ)}{\cos^{2} (θ)} = \frac{\sin^{2} (θ)}{\cos^{2} (θ)} + 1 = \tan^{2} (θ) + 1 \\ = & \frac{x}{a} + \sqrt{{(\frac{x}{a})}^{2} + 1} \dots x = a \tan (θ) よ り \tan (θ) = \frac{x}{a} \\ = & \frac{x}{a} + \sqrt{\frac{x^{2} + a^{2}}{a^{2}}} \\ = & \frac{x}{a} + \sqrt{\frac{1}{a^{2}} (x^{2} + a^{2})} \\ = & \frac{x}{a} + \frac{1}{a} \sqrt{x^{2} + a^{2}} \\ = & \frac{1}{a} (x + \sqrt{x^{2} + a^{2}}) \end{array}

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