Монгол Бодлогын Сан

Заавар: $\sin^2x=\dfrac{1-\cos2x}{2}$ ашиглан зэргийг бууруулж бод.

Бодолт: $$\displaystyle\int\cos^2x\,\,\mathrm{d}x=\int\dfrac{1-\cos2x}{2}\,\,\mathrm{d}x=\dfrac{x}{2}-\dfrac14\int \cos2x\,\mathrm{d}2x=\dfrac{x}{2}-\dfrac14\sin 2x+C.$$ Иймд $$\displaystyle\int_0^{\pi}\sin^2x\,\,\mathrm{d}x=\left.\left(\dfrac{x}{2}-\dfrac14\sin 2x\right)\right|_{0}^{\pi}=$$ $$=\dfrac{\pi}{2}-\dfrac14\sin(2\cdot \pi)-\dfrac{0}{2}+\dfrac14\sin(2\cdot 0)=\pi.$$