∫ (from 2 to 5) of 5/sqrt{x wrt x eching

$$\int _ { 2 } ^ { 5 } \frac { 5 } { \sqrt { x } } d x$$

Baholash

$$10\left(\sqrt{5}-\sqrt{2}\right)\approx 8,218544151$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \frac{5}{\sqrt{x}}\mathrm{d}x$$

\int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x yordamidan konstantani qavsdan tashqariga chiqaring.

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

\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.

$$10\sqrt{x}$$

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

$$10\times 5^{\frac{1}{2}}-10\times 2^{\frac{1}{2}}$$

Qisqartirish.

$$10\sqrt{5}-10\sqrt{2}$$