∫ (from 3 to 10) of -x^2+13x-30 wrt x eching

$$\int_{ 3 }^{ 10 } - { x }^{ 2 } +13x-30 d x$$

Baholash

$$\frac{343}{6}\approx 57,166666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int -x^{2}+13x-30\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int -x^{2}\mathrm{d}x+\int 13x\mathrm{d}x+\int -30\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$-\int x^{2}\mathrm{d}x+13\int x\mathrm{d}x+\int -30\mathrm{d}x$$

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.

$$-\frac{x^{3}}{3}+13\int x\mathrm{d}x+\int -30\mathrm{d}x$$

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

$$-\frac{x^{3}}{3}+\frac{13x^{2}}{2}+\int -30\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, -30 integralini toping.

$$-\frac{x^{3}}{3}+\frac{13x^{2}}{2}-30x$$

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

$$-\frac{10^{3}}{3}+\frac{13}{2}\times 10^{2}-30\times 10-\left(-\frac{3^{3}}{3}+\frac{13}{2}\times 3^{2}-30\times 3\right)$$

Qisqartirish.

$$\frac{343}{6}$$