Whakaoti i te ∫ (from 3 to 10) of -x^2+13x-30 wrt x

$$\int_{ 3 }^{ 10 } - { x }^{ 2 } +13x-30 d x$$

Aromātai

$$\frac{343}{6}\approx 57.166666667$$

Ngā Upane Otinga

Aromātaitia te tau tōpū tautuhi-kore i te tuatahi.

$$\int -x^{2}+13x-30\mathrm{d}x$$

Kōmitimititia te kīanga tapeke mā te kīanga.

$$\int -x^{2}\mathrm{d}x+\int 13x\mathrm{d}x+\int -30\mathrm{d}x$$

Whakatauwehea te pūmau i ēnei kīanga katoa.

$$-\int x^{2}\mathrm{d}x+13\int x\mathrm{d}x+\int -30\mathrm{d}x$$

Nā te mea \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} mō te k\neq -1, me whakakapi \int x^{2}\mathrm{d}x ki te \frac{x^{3}}{3}. Whakareatia -1 ki te \frac{x^{3}}{3}.

$$-\frac{x^{3}}{3}+13\int x\mathrm{d}x+\int -30\mathrm{d}x$$

Nā te mea \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} mō te k\neq -1, me whakakapi \int x\mathrm{d}x ki te \frac{x^{2}}{2}. Whakareatia 13 ki te \frac{x^{2}}{2}.

$$-\frac{x^{3}}{3}+\frac{13x^{2}}{2}+\int -30\mathrm{d}x$$

Kimihia te tau tōpū o -30 mā te whakamahi i te ture mō te ripanga o ngā tau tōpū pātahi \int a\mathrm{d}x=ax.

$$-\frac{x^{3}}{3}+\frac{13x^{2}}{2}-30x$$

Ko te tau tōpū tautuhi ko te pārōnaki kōaro o te kīanga i aromātaitia i te tepe tōrunga o te pāwhaitua, tangohia te pārōnaki kōaro i aromātaitia i te tepe tōraro o te pāwhaitua.

$$-\frac{10^{3}}{3}+\frac{13}{2}\times 10^{2}-30\times 10-\left(-\frac{3^{3}}{3}+\frac{13}{2}\times 3^{2}-30\times 3\right)$$

Whakarūnātia.

$$\frac{343}{6}$$