Whakaoti i te ∫ (x+1)(2x-1)(x+9) wrt x

$$\int ( x + 1 ) ( 2 x - 1 ) ( x + 9 ) d x$$

Aromātai

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x+С$$

Ngā Upane Otinga

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o x+1 ki ia tau o 2x-1.

$$\int \left(2x^{2}-x+2x-1\right)\left(x+9\right)\mathrm{d}x$$

Pahekotia te -x me 2x, ka x.

$$\int \left(2x^{2}+x-1\right)\left(x+9\right)\mathrm{d}x$$

Me hoatu te āhuatanga tohatoha mā te whakarea ia tau o 2x^{2}+x-1 ki ia tau o x+9.

$$\int 2x^{3}+18x^{2}+x^{2}+9x-x-9\mathrm{d}x$$

Pahekotia te 18x^{2} me x^{2}, ka 19x^{2}.

$$\int 2x^{3}+19x^{2}+9x-x-9\mathrm{d}x$$

Pahekotia te 9x me -x, ka 8x.

$$\int 2x^{3}+19x^{2}+8x-9\mathrm{d}x$$

Kōmitimititia te kīanga tapeke mā te kīanga.

$$\int 2x^{3}\mathrm{d}x+\int 19x^{2}\mathrm{d}x+\int 8x\mathrm{d}x+\int -9\mathrm{d}x$$

Whakatauwehea te pūmau i ēnei kīanga katoa.

$$2\int x^{3}\mathrm{d}x+19\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

Nā te mea \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} mō te k\neq -1, me whakakapi \int x^{3}\mathrm{d}x ki te \frac{x^{4}}{4}. Whakareatia 2 ki te \frac{x^{4}}{4}.

$$\frac{x^{4}}{2}+19\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}+\int -9\mathrm{d}x$$

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.

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x$$

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.

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x+С$$

Kimi Pārōnaki e ai ki x

$$\left(2x-1\right)\left(x+1\right)\left(x+9\right)$$