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

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

Baholash

$$\frac{10}{9}\approx 1,111111111$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int \frac{1}{x^{2}}\mathrm{d}x integralni -\frac{1}{x} bilan almashtiring.

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

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int \frac{1}{x^{3}}\mathrm{d}x integralni -\frac{1}{2x^{2}} bilan almashtiring. -1 ni -\frac{1}{2x^{2}} marotabaga ko'paytirish.

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

Qisqartirish.

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

Xos integral bu integral hisoblashning yuqori chegarasida hisoblangan ifodaning boshlangʻich holatidan chiqarib tashlagan holda integral hisoblashning quyi chegarasida hisoblangan ifodaning boshlangʻich holatidir.

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

Qisqartirish.

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