Réitigh ∫ (2/x^2+x+1) wrt x

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

Luacháil

$$\frac{x^{2}}{2}+x-\frac{2}{x}+С$$

Céimeanna Réitigh

Measc an tsuim téarma fá téarma.

$$\int \frac{2}{x^{2}}\mathrm{d}x+\int x\mathrm{d}x+\int 1\mathrm{d}x$$

Fág an leanúnach sna téarmaí as an áireamh.

$$2\int \frac{1}{x^{2}}\mathrm{d}x+\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$-\frac{2}{x}+\int x\mathrm{d}x+\int 1\mathrm{d}x$$

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}.

$$-\frac{2}{x}+\frac{x^{2}}{2}+\int 1\mathrm{d}x$$

Aimsigh suimeálaithe do 1 ag baint úsáid as an tábla do suimeálaithe coitianta riail\int a\mathrm{d}x=ax.

$$-\frac{2}{x}+\frac{x^{2}}{2}+x$$

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.

$$-\frac{2}{x}+\frac{x^{2}}{2}+x+С$$

Difreálaigh w.r.t. x

$$x+1+\frac{2}{x^{2}}$$