Réitigh ∫ (from - 3 to - 1) of (1/x^2-1/x^3) wrt x

$$\int _ { - 3 } ^ { - 1 } ( \frac { 1 } { x ^ { 2 } } - \frac { 1 } { x ^ { 3 } } ) d x$$

Luacháil

$$\frac{10}{9}\approx 1.111111111$$

Céimeanna Réitigh

Déan luacháil ar an suimeálaí éiginnte ar dtús.

$$\int \frac{1}{x^{2}}-\frac{1}{x^{3}}\mathrm{d}x$$

Measc an tsuim téarma fá téarma.

$$\int \frac{1}{x^{2}}\mathrm{d}x+\int -\frac{1}{x^{3}}\mathrm{d}x$$

Fág an leanúnach sna téarmaí as an áireamh.

$$\int \frac{1}{x^{2}}\mathrm{d}x-\int \frac{1}{x^{3}}\mathrm{d}x$$

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}.

$$-\frac{1}{x}-\int \frac{1}{x^{3}}\mathrm{d}x$$

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

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

Simpligh.

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

Is ionann suimeálaí cinnte agus frithdhíorthach an nath luacháilte ag teorainn uachtair na suimeála lúide an frithdhíorthach luacháilte ag teorainn íochtair na suimeála.

$$\left(\frac{1}{2}-\left(-1\right)\right)\left(-1\right)^{-2}-\left(\frac{1}{2}-\left(-3\right)\right)\left(-3\right)^{-2}$$

Simpligh.

$$\frac{10}{9}$$