sinの二乗の不定積分

sinの二乗の不定積分

$$\begin{array}{rcl}\int {\sin }^2\left(\theta \right)\mathrm{d}\theta & =& \int \frac{1}{2}-\frac{1}{2}\cos \left(2\theta \right)\mathrm{d}\theta \\ & & \cdots \cos \left(2\theta \right)={\cos }^2\left(\theta \right)-{\sin }^2\left(\theta \right)=(1-{\sin }^2\left(\theta \right))-{\sin }^2\left(\theta \right)=1-2{\sin }^2\left(\theta \right)\\ & & \cdots {\sin }^2\left(\theta \right)=\frac{1}{2}-\frac{1}{2}\cos \left(2\theta \right)\\ & =& \frac{1}{2}\int \mathrm{d}\theta -\frac{1}{2}\int \cos \left(2\theta \right)\mathrm{d}\theta \\ & =& \frac{1}{2}\theta -\frac{1}{2}\int \cos \left(2\theta \right)\mathrm{d}\theta \\ & =& \frac{1}{2}\theta -\frac{1}{2}\int \cos \left(\varphi \right)\frac{1}{2}\mathrm{d}\varphi \cdots \varphi =2\theta ,\frac{\mathrm{d}\varphi }{\mathrm{d}\theta }=2,\mathrm{d}\theta =\frac{1}{2}\mathrm{d}\varphi \\ & =& \frac{1}{2}\theta -\frac{1}{4}\int \cos \left(\varphi \right)\mathrm{d}\varphi \cdots \int cf(x)\mathrm{d}x=c\int f(x)\mathrm{d}x\\ & =& \frac{1}{2}\theta -\frac{1}{2}(\frac{1}{2}\sin \left(\varphi \right))\cdots \int \cos \left(x\right)\mathrm{d}x=\sin \left(x\right)+C(C:\mathrm{積}\mathrm{分}\mathrm{定}\mathrm{数})\\ & =& \frac{1}{2}\theta -\frac{1}{4}\sin \left(\varphi \right)\\ & =& \frac{1}{2}\theta -\frac{1}{4}\sin \left(2\theta \right)\\ & =& \frac{1}{2}\theta -\frac{1}{2}\sin \left(\theta \right)\cos \left(\theta \right)\\ & =& \frac{1}{2}\{\theta -\sin \left(\theta \right)\cos \left(\theta \right)\}+C\cdots C:\mathrm{積}\mathrm{分}\mathrm{定}\mathrm{数}\end{array}$$