Монгол Бодлогын Сан

Заавар:$$\int_a^b uv^\prime\,\mathrm{d}x=uv\bigg|_a^b-\int_a^b u^\prime v\,\mathrm{d}x$$ тодорхой интегралын хэсэгчлэн интегралчлах дүрмийг ашигла.

Бодолт:\begin{align*} \int_0^{\frac{\pi}2}&(x+1)\cos x\,\mathrm{d}x=\int_0^{\frac{\pi}2}(x+1)(\sin x)^\prime\,\mathrm{d}x\\ &=(x+1)\sin x\bigg|_0^{\frac{\pi}{2}}-\int_0^{\frac{\pi}{2}}\sin x\,\mathrm{d}x\\ &=(x+1)\sin x\bigg|_0^{\frac{\pi}{2}}+\cos x\bigg|_0^{\frac\pi2}\\ &=\left(\frac{\pi}{2}+1\right)\cdot\sin\dfrac{\pi}{2}-(0+1)\cdot\sin0+\cos\dfrac{\pi}{2}-\cos0\\ &=\dfrac{\pi}{2}+1-0+0-1=\dfrac{\pi}{2} \end{align*}