∫ (from 1 to 8) of (1/sqrt{x}-3x+5) wrt x= eching

$$\int _ { 1 } ^ { 8 } ( \frac { 1 } { \sqrt { x } } - 3 x + 5 ) d x =$$

Baholash

$$4\sqrt{2}-\frac{123}{2}\approx -55,843145751$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \frac{1}{\sqrt{x}}-3x+5\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int \frac{1}{\sqrt{x}}\mathrm{d}x+\int -3x\mathrm{d}x+\int 5\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int \frac{1}{\sqrt{x}}\mathrm{d}x-3\int x\mathrm{d}x+\int 5\mathrm{d}x$$

\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.

$$2\sqrt{x}-3\int x\mathrm{d}x+\int 5\mathrm{d}x$$

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. -3 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$2\sqrt{x}-\frac{3x^{2}}{2}+\int 5\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 5 integralini toping.

$$2\sqrt{x}-\frac{3x^{2}}{2}+5x$$

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

$$2\times 8^{\frac{1}{2}}-\frac{3}{2}\times 8^{2}+5\times 8-\left(2\times 1^{\frac{1}{2}}-\frac{3}{2}\times 1^{2}+5\times 1\right)$$

Qisqartirish.

$$4\sqrt{2}-\frac{123}{2}$$