Whakaoti i te ∫ (-x^2-4x-2)(-3x-7) wrt x

$$\int ( - x ^ { 2 } - 4 x - 2 ) ( - 3 x - 7 ) d x$$

Aromātai

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

Ngā Upane Otinga

Whakamahia te āhuatanga tuaritanga hei whakarea te -x^{2}-4x-2 ki te -3x-7 ka whakakotahi i ngā kupu rite.

$$\int -3\left(-x^{2}\right)x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

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

$$\int 3x^{2}x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

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.

$$\int 3x^{3}-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

Whakareatia te -7 ki te -1, ka 7.

$$\int 3x^{3}+7x^{2}+12x^{2}+34x+14\mathrm{d}x$$

Pahekotia te 7x^{2} me 12x^{2}, ka 19x^{2}.

$$\int 3x^{3}+19x^{2}+34x+14\mathrm{d}x$$

Kōmitimititia te kīanga tapeke mā te kīanga.

$$\int 3x^{3}\mathrm{d}x+\int 19x^{2}\mathrm{d}x+\int 34x\mathrm{d}x+\int 14\mathrm{d}x$$

Whakatauwehea te pūmau i ēnei kīanga katoa.

$$3\int x^{3}\mathrm{d}x+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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}.

$$\frac{3x^{4}}{4}+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+\int 14\mathrm{d}x$$

Kimihia te tau tōpū o 14 mā te whakamahi i te ture mō te ripanga o ngā tau tōpū pātahi \int a\mathrm{d}x=ax.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x$$

Mēnā ko F\left(x\right) he pārōnaki kōaro o f\left(x\right), kāti ko te huinga o ngā pārōnaki kōaro katoa o f\left(x\right) ka whakaaturia e F\left(x\right)+C. Nō reira, me tāpiri te pūmau o te whakatōpūtanga C\in \mathrm{R} ki te otinga.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

Kimi Pārōnaki e ai ki x

$$\left(3x+7\right)\left(x^{2}+4x+2\right)$$