∫ 5(x+2)-(x-1)(x+4)-6x eching

$$\int 5 ( x + 2 ) - ( x - 1 ) ( x + 4 ) - 6 x$$

Baholash

$$-\frac{x^{3}}{3}-2x^{2}+14x+С$$

Yechim qadamlari

5 ga x+2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 5x+10-\left(x-1\right)\left(x+4\right)-6x\mathrm{d}x$$

x-1 ifodaning har bir elementini x+4 ifodaning har bir elementiga ko‘paytirish orqali taqsimot qonuni xususiyatlarini qo‘llash mumkin.

$$\int 5x+10-\left(x^{2}+4x-x-4\right)-6x\mathrm{d}x$$

3x ni olish uchun 4x va -x ni birlashtirish.

$$\int 5x+10-\left(x^{2}+3x-4\right)-6x\mathrm{d}x$$

x^{2}+3x-4 teskarisini topish uchun har birining teskarisini toping.

$$\int 5x+10-x^{2}-3x-\left(-4\right)-6x\mathrm{d}x$$

-4 ning teskarisi 4 ga teng.

$$\int 5x+10-x^{2}-3x+4-6x\mathrm{d}x$$

2x ni olish uchun 5x va -3x ni birlashtirish.

$$\int 2x+10-x^{2}+4-6x\mathrm{d}x$$

14 olish uchun 10 va 4'ni qo'shing.

$$\int 2x+14-x^{2}-6x\mathrm{d}x$$

-4x ni olish uchun 2x va -6x ni birlashtirish.

$$\int -4x+14-x^{2}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int -4x\mathrm{d}x+\int 14\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$-4\int x\mathrm{d}x+\int 14\mathrm{d}x-\int x^{2}\mathrm{d}x$$

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

$$-2x^{2}+\int 14\mathrm{d}x-\int x^{2}\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 14 integralini toping.

$$-2x^{2}+14x-\int x^{2}\mathrm{d}x$$

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. -1 ni \frac{x^{3}}{3} marotabaga ko'paytirish.

$$-2x^{2}+14x-\frac{x^{3}}{3}$$

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.

$$-2x^{2}+14x-\frac{x^{3}}{3}+С$$

x ga nisbatan hosilani topish

$$14-4x-x^{2}$$