∫ (from 0 to pi) of sqrt{2x^3 wrt x eching

$$\int_{ 0 }^{ \pi } \sqrt{ 2 { x }^{ 3 } } d x$$

Baholash

$$\frac{2\sqrt{2}\pi ^{\frac{5}{2}}}{5}\approx 9,89577178$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \sqrt{2x^{3}}\mathrm{d}x$$

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

$$\sqrt{2}\int \sqrt{x^{3}}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{\frac{3}{2}}\mathrm{d}x integralni \frac{2x^{\frac{5}{2}}}{5} bilan almashtiring.

$$\sqrt{2}\times \frac{2x^{\frac{5}{2}}}{5}$$

Qisqartirish.

$$\frac{2\sqrt{2}x^{\frac{5}{2}}}{5}$$

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

$$\frac{2}{5}\times 2^{\frac{1}{2}}\pi ^{\frac{5}{2}}-\frac{2}{5}\times 2^{\frac{1}{2}}\times 0^{\frac{5}{2}}$$

Qisqartirish.

$$\frac{\left(2\pi \right)^{\frac{5}{2}}}{10}$$