f ( x ) = ∫ (from 0 to x) of (t^2-t)dt eching

$$f ( x ) = \int _ { 0 } ^ { x } ( t ^ { 2 } - t ) d t$$

Baholash

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

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int t^{2}\mathrm{d}t integralni \frac{t^{3}}{3} bilan almashtiring.

$$\frac{t^{3}}{3}-\int t\mathrm{d}t$$

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int t\mathrm{d}t integralni \frac{t^{2}}{2} bilan almashtiring. -1 ni \frac{t^{2}}{2} marotabaga ko'paytirish.

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

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

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

Qisqartirish.

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

x ga nisbatan hosilani topish

$$x\left(x-1\right)$$