∫ (from 0 to 3) of (5x-3)^2 wrt x= eching

$$\int _ { 0 } ^ { 3 } ( 5 x - 3 ) ^ { 2 } d x =$$

Baholash

$$117$$

Yechim qadamlari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(5x-3\right)^{2} kengaytirilishi uchun ishlating.

$$\int _{0}^{3}25x^{2}-30x+9\mathrm{d}x$$

Avval noaniq integralni baholang.

$$\int 25x^{2}-30x+9\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 25x^{2}\mathrm{d}x+\int -30x\mathrm{d}x+\int 9\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$25\int x^{2}\mathrm{d}x-30\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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. 25 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$\frac{25x^{3}}{3}-30\int x\mathrm{d}x+\int 9\mathrm{d}x$$

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. -30 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{25x^{3}}{3}-15x^{2}+\int 9\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 9 integralini toping.

$$\frac{25x^{3}}{3}-15x^{2}+9x$$

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

$$\frac{25}{3}\times 3^{3}-15\times 3^{2}+9\times 3-\left(\frac{25}{3}\times 0^{3}-15\times 0^{2}+9\times 0\right)$$

Qisqartirish.

$$117$$