∫ 2xleft(x^2+1right)^3 wrt x eching

$$\int{ 2x { \left( { x }^{ 2 } +1 \right) }^{ 3 } }d x$$

Baholash

$$\frac{x^{8}}{4}+x^{6}+\frac{3x^{4}}{2}+x^{2}+С$$

Yechim qadamlari

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

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

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 3 ni ko‘paytirib, 6 ni oling.

$$\int 2x\left(x^{6}+3\left(x^{2}\right)^{2}+3x^{2}+1\right)\mathrm{d}x$$

Daraja ko‘rsatkichini boshqa ko‘rsatkichga oshirish uchun, darajalarini ko‘paytiring. 2 va 2 ni ko‘paytirib, 4 ni oling.

$$\int 2x\left(x^{6}+3x^{4}+3x^{2}+1\right)\mathrm{d}x$$

2x ga x^{6}+3x^{4}+3x^{2}+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int 2x^{7}+6x^{5}+6x^{3}+2x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{7}\mathrm{d}x+\int 6x^{5}\mathrm{d}x+\int 6x^{3}\mathrm{d}x+\int 2x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{7}\mathrm{d}x+6\int x^{5}\mathrm{d}x+6\int x^{3}\mathrm{d}x+2\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{7}\mathrm{d}x with \frac{x^{8}}{8}. 2 ni \frac{x^{8}}{8} marotabaga ko'paytirish.

$$\frac{x^{8}}{4}+6\int x^{5}\mathrm{d}x+6\int x^{3}\mathrm{d}x+2\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{5}\mathrm{d}x with \frac{x^{6}}{6}. 6 ni \frac{x^{6}}{6} marotabaga ko'paytirish.

$$\frac{x^{8}}{4}+x^{6}+6\int x^{3}\mathrm{d}x+2\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. 6 ni \frac{x^{4}}{4} marotabaga ko'paytirish.

$$\frac{x^{8}}{4}+x^{6}+\frac{3x^{4}}{2}+2\int x\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. 2 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$\frac{x^{8}}{4}+x^{6}+\frac{3x^{4}}{2}+x^{2}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$x^{2}+\frac{3x^{4}}{2}+x^{6}+\frac{x^{8}}{4}+С$$

x ga nisbatan hosilani topish

$$2x\left(x^{2}+1\right)^{3}$$