∫ (from 1 to 5) of (15t^3-135t^2+225t)dt= eching

$$\int _ { 1 } ^ { 5 } ( 15 t ^ { 3 } - 135 t ^ { 2 } + 225 t ) d t =$$

Baholash

$$-540$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int 15t^{3}-135t^{2}+225t\mathrm{d}t$$

Summani muddatma-muddat integratsiya qiling.

$$\int 15t^{3}\mathrm{d}t+\int -135t^{2}\mathrm{d}t+\int 225t\mathrm{d}t$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$15\int t^{3}\mathrm{d}t-135\int t^{2}\mathrm{d}t+225\int t\mathrm{d}t$$

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int t^{3}\mathrm{d}t integralni \frac{t^{4}}{4} bilan almashtiring. 15 ni \frac{t^{4}}{4} marotabaga ko'paytirish.

$$\frac{15t^{4}}{4}-135\int t^{2}\mathrm{d}t+225\int t\mathrm{d}t$$

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. -135 ni \frac{t^{3}}{3} marotabaga ko'paytirish.

$$\frac{15t^{4}}{4}-45t^{3}+225\int t\mathrm{d}t$$

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

$$\frac{15t^{4}}{4}-45t^{3}+\frac{225t^{2}}{2}$$

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

$$\frac{15}{4}\times 5^{4}-45\times 5^{3}+\frac{225}{2}\times 5^{2}-\left(\frac{15}{4}\times 1^{4}-45\times 1^{3}+\frac{225}{2}\times 1^{2}\right)$$

Qisqartirish.

$$-540$$