Réitigh ∫ x(x+1)^2

$$\int x ( x + 1 ) ^ { 2 }$$

Luacháil

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

Céimeanna Réitigh

Úsáid an teoirim dhéthéarmach \left(a+b\right)^{2}=a^{2}+2ab+b^{2} chun \left(x+1\right)^{2} a leathnú.

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

Úsáid an t-airí dáileach chun x a mhéadú faoi x^{2}+2x+1.

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

Measc an tsuim téarma fá téarma.

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

Fág an leanúnach sna téarmaí as an áireamh.

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

Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x^{3}\mathrm{d}x le \frac{x^{4}}{4}.

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

Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x^{2}\mathrm{d}x le \frac{x^{3}}{3}. Méadaigh 2 faoi \frac{x^{3}}{3}.

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

Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x\mathrm{d}x le \frac{x^{2}}{2}.

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

Simpligh.

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

Má tá F\left(x\right) mar frithdhíorthach do f\left(x\right), beidh tacar do frithdhíorthach uile do f\left(x\right) a thabhairt ag F\left(x\right)+C. Mar sin de, cur an comhtháthú leanúnach C\in \mathrm{R} don toradh.

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

Difreálaigh w.r.t. x

$$x\left(x+1\right)^{2}$$