∫ (-x^2-4x-2)(-3x-7) wrt x eching

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

Baholash

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

Yechim qadamlari

-x^{2}-4x-2 ga -3x-7 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$\int -3\left(-x^{2}\right)x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

3 hosil qilish uchun -3 va -1 ni ko'paytirish.

$$\int 3x^{2}x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 2 va 1 ni qo‘shib, 3 ni oling.

$$\int 3x^{3}-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

7 hosil qilish uchun -7 va -1 ni ko'paytirish.

$$\int 3x^{3}+7x^{2}+12x^{2}+34x+14\mathrm{d}x$$

19x^{2} ni olish uchun 7x^{2} va 12x^{2} ni birlashtirish.

$$\int 3x^{3}+19x^{2}+34x+14\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 3x^{3}\mathrm{d}x+\int 19x^{2}\mathrm{d}x+\int 34x\mathrm{d}x+\int 14\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$3\int x^{3}\mathrm{d}x+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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.

$$\frac{3x^{4}}{4}+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+\int 14\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x$$

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.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

x ga nisbatan hosilani topish

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