∫ (from 0 to frac { 1) of {2}}sqrt{2x wrt x eching

$$\int _ { 0 } ^ { \frac { 1 } { 2 } } \sqrt { 2 x } d x$$

Baholash

$$\frac{1}{3}\approx 0,333333333$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

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

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

\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.

$$\sqrt{2}\times \frac{2x^{\frac{3}{2}}}{3}$$

Qisqartirish.

$$\frac{2\sqrt{2}x^{\frac{3}{2}}}{3}$$

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

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

Qisqartirish.

$$\frac{1}{3}$$