∫ (4x^7+4x+4)(28x^6+4) wrt x eching

$$\int ( 4 x ^ { 7 } + 4 x + 4 ) ( 28 x ^ { 6 } + 4 ) d x$$

Baholash

$$8x^{14}+16x^{8}+16x^{7}+8x^{2}+16x+С$$

Yechim qadamlari

4x^{7}+4x+4 ga 28x^{6}+4 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\int 112x^{13}+128x^{7}+16x+112x^{6}+16\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 112x^{13}\mathrm{d}x+\int 128x^{7}\mathrm{d}x+\int 16x\mathrm{d}x+\int 112x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$112\int x^{13}\mathrm{d}x+128\int x^{7}\mathrm{d}x+16\int x\mathrm{d}x+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{13}\mathrm{d}x integralni \frac{x^{14}}{14} bilan almashtiring. 112 ni \frac{x^{14}}{14} marotabaga ko'paytirish.

$$8x^{14}+128\int x^{7}\mathrm{d}x+16\int x\mathrm{d}x+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{7}\mathrm{d}x integralni \frac{x^{8}}{8} bilan almashtiring. 128 ni \frac{x^{8}}{8} marotabaga ko'paytirish.

$$8x^{14}+16x^{8}+16\int x\mathrm{d}x+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring. 16 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$8x^{14}+16x^{8}+8x^{2}+112\int x^{6}\mathrm{d}x+\int 16\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{6}\mathrm{d}x integralni \frac{x^{7}}{7} bilan almashtiring. 112 ni \frac{x^{7}}{7} marotabaga ko'paytirish.

$$8x^{14}+16x^{8}+8x^{2}+16x^{7}+\int 16\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 16 integralini toping.

$$8x^{14}+16x^{8}+8x^{2}+16x^{7}+16x$$

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.

$$8x^{14}+16x^{8}+16x^{7}+8x^{2}+16x+С$$

x ga nisbatan hosilani topish

$$16\left(7x^{6}+1\right)\left(x^{7}+x+1\right)$$