∫ (from 3 to 50) of x(x+8) wrt x eching

$$\int _ { 3 } ^ { 50 } x ( x + 8 ) d x$$

Baholash

$$\frac{154865}{3}\approx 51621,666666667$$

Yechim qadamlari

x ga x+8 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int _{3}^{50}x^{2}+8x\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int x^{2}+8x\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{2}\mathrm{d}x+\int 8x\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x$$

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.

$$\frac{x^{3}}{3}+8\int x\mathrm{d}x$$

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. 8 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

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

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

$$\frac{50^{3}}{3}+4\times 50^{2}-\left(\frac{3^{3}}{3}+4\times 3^{2}\right)$$

Qisqartirish.

$$\frac{154865}{3}$$