∫ (7x^2-3x^3+4x^5) wrt x eching

$$\int ( 7 x ^ { 2 } - 3 x ^ { 3 } + 4 x ^ { 5 } ) d x$$

Baholash

$$\frac{2x^{6}}{3}-\frac{3x^{4}}{4}+\frac{7x^{3}}{3}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 7x^{2}\mathrm{d}x+\int -3x^{3}\mathrm{d}x+\int 4x^{5}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$7\int x^{2}\mathrm{d}x-3\int x^{3}\mathrm{d}x+4\int x^{5}\mathrm{d}x$$

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

$$\frac{7x^{3}}{3}-3\int x^{3}\mathrm{d}x+4\int x^{5}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{3}\mathrm{d}x integralni \frac{x^{4}}{4} bilan almashtiring. -3 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

$$\frac{7x^{3}}{3}-\frac{3x^{4}}{4}+4\int x^{5}\mathrm{d}x$$

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. 4 ni \frac{x^{6}}{6} marotabaga ko'paytirish.

$$\frac{7x^{3}}{3}-\frac{3x^{4}}{4}+\frac{2x^{6}}{3}$$

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.

$$\frac{7x^{3}}{3}-\frac{3x^{4}}{4}+\frac{2x^{6}}{3}+С$$

x ga nisbatan hosilani topish

$$x^{2}\left(4x^{3}-3x+7\right)$$