f ( x ) = ∫ (from 0 to x) of (t^3+2t^2+1)dt eching

$$f ( x ) = \int _ { 0 } ^ { x } ( t ^ { 3 } + 2 t ^ { 2 } + 1 ) d t$$

Baholash

$$\frac{x^{4}}{4}+\frac{2x^{3}}{3}+x$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int t^{3}+2t^{2}+1\mathrm{d}t$$

Summani muddatma-muddat integratsiya qiling.

$$\int t^{3}\mathrm{d}t+\int 2t^{2}\mathrm{d}t+\int 1\mathrm{d}t$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int t^{3}\mathrm{d}t+2\int t^{2}\mathrm{d}t+\int 1\mathrm{d}t$$

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int t^{3}\mathrm{d}t integralni \frac{t^{4}}{4} bilan almashtiring.

$$\frac{t^{4}}{4}+2\int t^{2}\mathrm{d}t+\int 1\mathrm{d}t$$

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. 2 ni \frac{t^{3}}{3} marotabaga ko'paytirish.

$$\frac{t^{4}}{4}+\frac{2t^{3}}{3}+\int 1\mathrm{d}t$$

\int a\mathrm{d}t=at umumiy integrallar qoidasi jadvalidan foydalanib, 1 integralini toping.

$$\frac{t^{4}}{4}+\frac{2t^{3}}{3}+t$$

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

$$\frac{x^{4}}{4}+\frac{2}{3}x^{3}+x-\left(\frac{0^{4}}{4}+\frac{2}{3}\times 0^{3}+0\right)$$

Qisqartirish.

$$\frac{x^{4}}{4}+\frac{2x^{3}}{3}+x$$

x ga nisbatan hosilani topish

$$x^{3}+2x^{2}+1$$