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| 44行目: | 44行目: | ||
\begin{aligned} | \begin{aligned} | ||
N \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dfrac{\alpha}{\alpha^{2} \left( 1 + \tan^{2}{\theta} \right) } \dfrac{\alpha}{\cos^{2}{\theta}} \, d \theta \\ | N \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dfrac{\alpha}{\alpha^{2} \left( 1 + \tan^{2}{\theta} \right) } \dfrac{\alpha}{\cos^{2}{\theta}} \, d \theta \\ | ||
&= N \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \dfrac{1}{\cos^{2}{\theta} + \sin^{2}{\theta}} \, d \theta \qquad \because \tan^{2}{\theta} = \dfrac{\sin^{2}{\theta}}{\cos^{2}{\theta}}\\ | |||
&= N \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \, d \theta \qquad \because \sin^{2}{\theta} + \cos^{2}{\theta} = 1 \\ | |||
&= N \left[ \theta \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \\ | &= N \left[ \theta \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \\ | ||
&= N \pi | &= N \pi | ||