Монгол Бодлогын Сан

Заавар: $$\displaystyle\int_a^b f(x)\,\mathrm{d}x=\int_a^c f(x)\,\mathrm{d}x+\int_c^b f(x)\,\mathrm{d}x$$ шугаман чанар ашиглан бод.

Бодолт: $$\begin{align*} \text{Инт.}&=\int_{0}^4|-x^2+5x-6|\,\mathrm{d}x=\\ &=\int_{0}^2|-x^2+5x-6|\,\mathrm{d}x+{} & & \color{red}{\leftarrow-x^2+5x-6<0}\\ &\phantom{{}={}}+\int_{2}^3|-x^2+5x-6|\,\mathrm{d}x+{} & & \color{red}{\leftarrow-x^2+5x-6>0}\\ &\phantom{{}={}}+\int_{3}^4|-x^2+5x-6|\,\mathrm{d}x={} & & \color{red}{\leftarrow-x^2+5x-6<0}\\ &=-\int_{0}^2(-x^2+5x-6)\,\mathrm{d}x+{}\\ &\phantom{{}={}}+\int_{2}^3(-x^2+5x-6)\,\mathrm{d}x-{}\\ &\phantom{{}={}}-\int_{3}^4(-x^2+5x-6)\,\mathrm{d}x={}\\ &=-\left(-\dfrac{x^3}{3}+\dfrac{5x^2}{2}-6x\right)\Bigg|_0^2+{}\\ &\phantom{{}={}}+\left(-\dfrac{x^3}{3}+\dfrac{5x^2}{2}-6x\right)\Bigg|_2^3-{}\\ &=-\left(-\dfrac{x^3}{3}+\dfrac{5x^2}{2}-6x\right)\Bigg|_3^4={}\\ &=-\left(-\dfrac83+10-12\right)+0+{}\\ &\phantom{{}={}}+\left(-9+\dfrac{45}{2}-18\right)-\left(-\dfrac83+10-12\right)-{}\\ &\phantom{{}={}}-\left(-\dfrac{64}{3}+40-24\right)+\left(-9+\dfrac{45}{2}-18\right)={}\\ &=\dfrac{14}{3}+0-\dfrac{9}{2}+\dfrac{14}{3}+\dfrac{16}{3}-\dfrac{9}{2}=\dfrac{17}{3}=5\dfrac23 \end{align*}$$