∫ x^2(x^3+2x^2-5) wrt x eching

$$\int x ^ { 2 } ( x ^ { 3 } + 2 x ^ { 2 } - 5 ) d x$$

Baholash

$$\frac{x^{6}}{6}+\frac{2x^{5}}{5}-\frac{5x^{3}}{3}+С$$

Yechim qadamlari

x^{2} ga x^{3}+2x^{2}-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int x^{5}+2x^{4}-5x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int x^{5}\mathrm{d}x+\int 2x^{4}\mathrm{d}x+\int -5x^{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int x^{5}\mathrm{d}x+2\int x^{4}\mathrm{d}x-5\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{5}\mathrm{d}x integralni \frac{x^{6}}{6} bilan almashtiring.

$$\frac{x^{6}}{6}+2\int x^{4}\mathrm{d}x-5\int x^{2}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{4}\mathrm{d}x integralni \frac{x^{5}}{5} bilan almashtiring. 2 ni \frac{x^{5}}{5} marotabaga ko'paytirish.

$$\frac{x^{6}}{6}+\frac{2x^{5}}{5}-5\int x^{2}\mathrm{d}x$$

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. -5 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$\frac{x^{6}}{6}+\frac{2x^{5}}{5}-\frac{5x^{3}}{3}$$

Агар F\left(x\right)f\left(x\right) ning dastlabki holati boʻlsa, u holatda f\left(x\right) ning barcha dastlabki holatlari toʻplami F\left(x\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

$$\frac{x^{6}}{6}+\frac{2x^{5}}{5}-\frac{5x^{3}}{3}+С$$

x ga nisbatan hosilani topish

$$x^{2}\left(x^{3}+2x^{2}-5\right)$$