∫ (from 1 to 2) of (1/sqrt{x}-x) wrt x eching

$$\int _ { 1 } ^ { 2 } ( \frac { 1 } { \sqrt { x } } - x ) d x$$

Baholash

$$2\sqrt{2}-\frac{7}{2}\approx -0,671572875$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \frac{1}{\sqrt{x}}-x\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int \frac{1}{\sqrt{x}}\mathrm{d}x+\int -x\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int \frac{1}{\sqrt{x}}\mathrm{d}x-\int x\mathrm{d}x$$

\frac{1}{\sqrt{x}} ni x^{-\frac{1}{2}} sifatida qaytadan yozish. k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{-\frac{1}{2}}\mathrm{d}x integralni \frac{x^{\frac{1}{2}}}{\frac{1}{2}} bilan almashtiring. Soddalashtiring va eksponensialdan ildiz shaklga aylantiring.

$$2\sqrt{x}-\int x\mathrm{d}x$$

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. -1 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$2\sqrt{x}-\frac{x^{2}}{2}$$

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

$$2\times 2^{\frac{1}{2}}-\frac{2^{2}}{2}-\left(2\times 1^{\frac{1}{2}}-\frac{1^{2}}{2}\right)$$

Qisqartirish.

$$2\sqrt{2}-\frac{7}{2}$$