Whakaoti i te ∫ (from 0 to 3) of (-3x^2+12x+2)(-x) wrt x

$$\int_{ 0 }^{ 3 } (-3 { x }^{ 2 } +12x+2)(-x) d x$$

Aromātai

$$-\frac{225}{4}=-56.25$$

Ngā Upane Otinga

Whakamahia te āhuatanga tohatoha hei whakarea te -3x^{2}+12x+2 ki te -x.

$$\int _{0}^{3}-3x^{2}\left(-x\right)+12x\left(-x\right)+2\left(-x\right)\mathrm{d}x$$

Whakareatia te -3 ki te -1, ka 3.

$$\int _{0}^{3}3x^{2}x+12x\left(-x\right)+2\left(-x\right)\mathrm{d}x$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 2 me te 1 kia riro ai te 3.

$$\int _{0}^{3}3x^{3}+12x\left(-x\right)+2\left(-x\right)\mathrm{d}x$$

Whakareatia te x ki te x, ka x^{2}.

$$\int _{0}^{3}3x^{3}+12x^{2}\left(-1\right)+2\left(-1\right)x\mathrm{d}x$$

Whakareatia te 12 ki te -1, ka -12.

$$\int _{0}^{3}3x^{3}-12x^{2}+2\left(-1\right)x\mathrm{d}x$$

Whakareatia te 2 ki te -1, ka -2.

$$\int _{0}^{3}3x^{3}-12x^{2}-2x\mathrm{d}x$$

Aromātaitia te tau tōpū tautuhi-kore i te tuatahi.

$$\int 3x^{3}-12x^{2}-2x\mathrm{d}x$$

Kōmitimititia te kīanga tapeke mā te kīanga.

$$\int 3x^{3}\mathrm{d}x+\int -12x^{2}\mathrm{d}x+\int -2x\mathrm{d}x$$

Whakatauwehea te pūmau i ēnei kīanga katoa.

$$3\int x^{3}\mathrm{d}x-12\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x$$

Nā te mea \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} mō te k\neq -1, me whakakapi \int x^{3}\mathrm{d}x ki te \frac{x^{4}}{4}. Whakareatia 3 ki te \frac{x^{4}}{4}.

$$\frac{3x^{4}}{4}-12\int x^{2}\mathrm{d}x-2\int x\mathrm{d}x$$

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 -12 ki te \frac{x^{3}}{3}.

$$\frac{3x^{4}}{4}-4x^{3}-2\int x\mathrm{d}x$$

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 -2 ki te \frac{x^{2}}{2}.

$$\frac{3x^{4}}{4}-4x^{3}-x^{2}$$

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.

$$\frac{3}{4}\times 3^{4}-4\times 3^{3}-3^{2}-\left(\frac{3}{4}\times 0^{4}-4\times 0^{3}-0^{2}\right)$$

Whakarūnātia.

$$-\frac{225}{4}$$