∫ x(2tx^2)^2 wrt x eching

$$\int x ( 2 t x ^ { 2 } ) ^ { 2 } d x$$

Baholash

$$\frac{2t^{2}x^{6}}{3}+С$$

Yechim qadamlari

\left(2tx^{2}\right)^{2} ni kengaytirish.

$$\int x\times 2^{2}t^{2}\left(x^{2}\right)^{2}\mathrm{d}x$$

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

$$\int x\times 2^{2}t^{2}x^{4}\mathrm{d}x$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$\int x\times 4t^{2}x^{4}\mathrm{d}x$$

Ayni asosning daraja ko‘rsatkichlarini ko‘paytirish uchun ularning darajalarini qo‘shing. 1 va 4 ni qo‘shib, 5 ni oling.

$$\int x^{5}\times 4t^{2}\mathrm{d}x$$

\int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x yordamidan konstantani qavsdan tashqariga chiqaring.

$$4t^{2}\int x^{5}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{5}\mathrm{d}x integralni \frac{x^{6}}{6} bilan almashtiring.

$$4t^{2}\times \frac{x^{6}}{6}$$

Qisqartirish.

$$\frac{2t^{2}x^{6}}{3}$$

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.

$$\frac{2t^{2}x^{6}}{3}+С$$

x ga nisbatan hosilani topish

$$4t^{2}x^{5}$$