∫ (x-1)^{3}+(x-1)^{2}-x+x(4-x)(4+x)+(8-x-x^2)^2+x^2(17-x^2) eching

$$\int ( x - 1 ) ^ { 3 } + ( x - 1 ) ^ { 2 } - x + x ( 4 - x ) ( 4 + x ) + ( 8 - x - x ^ { 2 } ) ^ { 2 } + x ^ { 2 } ( 17 - x ^ { 2 } )$$

Baholash

$$\frac{x^{4}}{2}+64x+С$$

Yechim qadamlari

\left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} binom teoremasini \left(x-1\right)^{3} kengaytirilishi uchun ishlating.

$$\int x^{3}-3x^{2}+3x-1+\left(x-1\right)^{2}-x+x\left(4-x\right)\left(4+x\right)+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-1\right)^{2} kengaytirilishi uchun ishlating.

$$\int x^{3}-3x^{2}+3x-1+x^{2}-2x+1-x+x\left(4-x\right)\left(4+x\right)+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

-2x^{2} ni olish uchun -3x^{2} va x^{2} ni birlashtirish.

$$\int x^{3}-2x^{2}+3x-1-2x+1-x+x\left(4-x\right)\left(4+x\right)+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

x ni olish uchun 3x va -2x ni birlashtirish.

$$\int x^{3}-2x^{2}+x-1+1-x+x\left(4-x\right)\left(4+x\right)+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

0 olish uchun -1 va 1'ni qo'shing.

$$\int x^{3}-2x^{2}+x-x+x\left(4-x\right)\left(4+x\right)+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

x ga 4-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int x^{3}-2x^{2}+x-x+\left(4x-x^{2}\right)\left(4+x\right)+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

4x-x^{2} ga 4+x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\int x^{3}-2x^{2}+x-x+16x-x^{3}+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

17x ni olish uchun x va 16x ni birlashtirish.

$$\int x^{3}-2x^{2}+17x-x-x^{3}+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

0 ni olish uchun x^{3} va -x^{3} ni birlashtirish.

$$\int -2x^{2}+17x-x+\left(8-x-x^{2}\right)^{2}+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

8-x-x^{2} kvadratini chiqarish.

$$\int -2x^{2}+17x-x+x^{4}+2x^{3}-15x^{2}-16x+64+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

-17x^{2} ni olish uchun -2x^{2} va -15x^{2} ni birlashtirish.

$$\int -17x^{2}+17x-x+x^{4}+2x^{3}-16x+64+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

x ni olish uchun 17x va -16x ni birlashtirish.

$$\int -17x^{2}+x-x+x^{4}+2x^{3}+64+x^{2}\left(17-x^{2}\right)\mathrm{d}x$$

x^{2} ga 17-x^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int -17x^{2}+x-x+x^{4}+2x^{3}+64+17x^{2}-x^{4}\mathrm{d}x$$

0 ni olish uchun -17x^{2} va 17x^{2} ni birlashtirish.

$$\int x-x+x^{4}+2x^{3}+64-x^{4}\mathrm{d}x$$

0 ni olish uchun x^{4} va -x^{4} ni birlashtirish.

$$\int x-x+2x^{3}+64\mathrm{d}x$$

0 ni olish uchun x va -x ni birlashtirish.

$$\int 2x^{3}+64\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 2x^{3}\mathrm{d}x+\int 64\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$2\int x^{3}\mathrm{d}x+\int 64\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+\int 64\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 64 integralini toping.

$$\frac{x^{4}}{2}+64x$$

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.

$$64x+\frac{x^{4}}{2}+С$$

x ga nisbatan hosilani topish

$$2\left(x^{3}+32\right)$$