Whakaoti i te ∫ (from 0 to 1) of (2x+3)^2 wrt x

$$\int _ { 0 } ^ { 1 } ( 2 x + 3 ) ^ { 2 } d x$$

Aromātai

$$\frac{49}{3}\approx 16.333333333$$

Ngā Upane Otinga

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2x+3\right)^{2}.

$$\int _{0}^{1}4x^{2}+12x+9\mathrm{d}x$$

Aromātaitia te tau tōpū tautuhi-kore i te tuatahi.

$$\int 4x^{2}+12x+9\mathrm{d}x$$

Kōmitimititia te kīanga tapeke mā te kīanga.

$$\int 4x^{2}\mathrm{d}x+\int 12x\mathrm{d}x+\int 9\mathrm{d}x$$

Whakatauwehea te pūmau i ēnei kīanga katoa.

$$4\int x^{2}\mathrm{d}x+12\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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 4 ki te \frac{x^{3}}{3}.

$$\frac{4x^{3}}{3}+12\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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

$$\frac{4x^{3}}{3}+6x^{2}+\int 9\mathrm{d}x$$

Kimihia te tau tōpū o 9 mā te whakamahi i te ture mō te ripanga o ngā tau tōpū pātahi \int a\mathrm{d}x=ax.

$$\frac{4x^{3}}{3}+6x^{2}+9x$$

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.

$$\frac{4}{3}\times 1^{3}+6\times 1^{2}+9\times 1-\left(\frac{4}{3}\times 0^{3}+6\times 0^{2}+9\times 0\right)$$

Whakarūnātia.

$$\frac{49}{3}$$