∫ (from 0 to 2 cos x) of r-r^2dr eching

$$\int _ { 0 } ^ { 2 \cos x } r - r ^ { 2 } d r$$

Baholash

$$\frac{2\left(-4\cos(x)+3\right)\left(\cos(x)\right)^{2}}{3}$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int r-r^{2}\mathrm{d}r$$

Summani muddatma-muddat integratsiya qiling.

$$\int r\mathrm{d}r+\int -r^{2}\mathrm{d}r$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int r\mathrm{d}r-\int r^{2}\mathrm{d}r$$

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

$$\frac{r^{2}}{2}-\int r^{2}\mathrm{d}r$$

k\neq -1 uchun integral \int r^{k}\mathrm{d}r=\frac{r^{k+1}}{k+1} boʻlgani uchun, \int r^{2}\mathrm{d}r integralni \frac{r^{3}}{3} bilan almashtiring. -1 ni \frac{r^{3}}{3} marotabaga ko'paytirish.

$$\frac{r^{2}}{2}-\frac{r^{3}}{3}$$

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

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

Qisqartirish.

$$\left(\cos(x)\right)^{2}\left(2-\frac{8\cos(x)}{3}\right)$$

x ga nisbatan hosilani topish

$$2\sin(2x)\left(2\cos(x)-1\right)$$