∫ (2x^5+3/x+1/x^9) wrt x eching

$$\int ( 2 x ^ { 5 } + \frac { 3 } { x } + \frac { 1 } { x ^ { 9 } } ) d x$$

Baholash

$$3\ln(|x|)+\frac{x^{6}}{3}-\frac{1}{8x^{8}}+С$$

Yechim qadamlari

Summani muddatma-muddat integratsiya qiling.

$$\int 2x^{5}\mathrm{d}x+\int \frac{3}{x}\mathrm{d}x+\int \frac{1}{x^{9}}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int x^{5}\mathrm{d}x+3\int \frac{1}{x}\mathrm{d}x+\int \frac{1}{x^{9}}\mathrm{d}x$$

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

$$\frac{x^{6}}{3}+3\int \frac{1}{x}\mathrm{d}x+\int \frac{1}{x^{9}}\mathrm{d}x$$

Natijani olish uchun umumiy integrallar jadvalidagi \int \frac{1}{x}\mathrm{d}x=\ln(|x|) integralidan foydalaning.

$$\frac{x^{6}}{3}+3\ln(|x|)+\int \frac{1}{x^{9}}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int \frac{1}{x^{9}}\mathrm{d}x integralni -\frac{1}{8x^{8}} bilan almashtiring.

$$\frac{x^{6}}{3}+3\ln(|x|)-\frac{1}{8x^{8}}$$

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.

$$\frac{x^{6}}{3}+3\ln(|x|)-\frac{1}{8x^{8}}+С$$

x ga nisbatan hosilani topish

$$2x^{5}+\frac{3}{x}+\frac{1}{x^{9}}$$