∫ (from 1 to 3) of (3x^2+4/x^2) wrt x eching

$$\int _ { 1 } ^ { 3 } ( 3 x ^ { 2 } + \frac { 4 } { x ^ { 2 } } ) d x$$

Baholash

$$\frac{86}{3}\approx 28,666666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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

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

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int \frac{1}{x^{2}}\mathrm{d}x integralni -\frac{1}{x} bilan almashtiring. 4 ni -\frac{1}{x} marotabaga ko'paytirish.

$$x^{3}-\frac{4}{x}$$

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

$$3^{3}-4\times 3^{-1}-\left(1^{3}-4\times 1^{-1}\right)$$

Qisqartirish.

$$\frac{86}{3}$$