∫ (from 0 to 1) of 2^x+x^2 wrt x eching

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

Baholash

$$\frac{1}{\ln(2)}+\frac{1}{3}\approx 1,776028374$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int 2^{x}+x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 2^{x}\mathrm{d}x+\int x^{2}\mathrm{d}x$$

Natijani olish uchun umumiy integrallar jadvalidagi \int x^{k}\mathrm{d}k=\frac{x^{k}}{\ln(x)} integralidan foydalaning.

$$\frac{2^{x}}{\ln(2)}+\int x^{2}\mathrm{d}x$$

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

$$\frac{2^{x}}{\ln(2)}+\frac{x^{3}}{3}$$

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

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

Qisqartirish.

$$\frac{1}{3}+\frac{1}{\ln(2)}$$