∫ (from 0 to 16) of sqrt{x}+sqrt{y-4} eching

$$\int _ { 0 } ^ { 16 } \sqrt { x } + \sqrt { y - 4 }$$

Baholash

$$16\sqrt{y-4}+\frac{128}{3}$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \sqrt{x}+\sqrt{y-4}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int \sqrt{x}\mathrm{d}x+\int \sqrt{y-4}\mathrm{d}x$$

\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{3}{2}}}{\frac{3}{2}} bilan almashtiring. Qisqartirish.

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

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, \sqrt{y-4} integralini toping.

$$\frac{2x^{\frac{3}{2}}}{3}+\sqrt{y-4}x$$

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

$$\frac{2}{3}\times 16^{\frac{3}{2}}+\left(y-4\right)^{\frac{1}{2}}\times 16-\left(\frac{2}{3}\times 0^{\frac{3}{2}}+\left(y-4\right)^{\frac{1}{2}}\times 0\right)$$

Qisqartirish.

$$\frac{128}{3}+16\sqrt{y-4}$$

y ga nisbatan hosilani topish

$$\frac{8}{\sqrt{y-4}}$$