∫ (from 0 to 5) of (x-x^2) wrt x eching

$$\int _ { 0 } ^ { 5 } ( x - x ^ { 2 } ) d x$$

Baholash

$$-\frac{175}{6}\approx -29,166666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int x-x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x\mathrm{d}x-\int x^{2}\mathrm{d}x$$

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

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

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

$$\frac{x^{2}}{2}-\frac{x^{3}}{3}$$

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

$$\frac{5^{2}}{2}-\frac{5^{3}}{3}-\left(\frac{0^{2}}{2}-\frac{0^{3}}{3}\right)$$

Qisqartirish.

$$-\frac{175}{6}$$