∫ (from 0 to 1) of 3x^2+3x+sin(x) wrt x eching

$$\int_{ 0 }^{ 1 } 3 { x }^{ 2 } +3x+ \sin ( x ) d x$$

Baholash

$$\frac{7-2\cos(1)}{2}\approx 2,959697694$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int 3x^{2}+3x+\sin(x)\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int 3x^{2}\mathrm{d}x+\int 3x\mathrm{d}x+\int \sin(x)\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$3\int x^{2}\mathrm{d}x+3\int x\mathrm{d}x+\int \sin(x)\mathrm{d}x$$

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

$$x^{3}+3\int x\mathrm{d}x+\int \sin(x)\mathrm{d}x$$

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

$$x^{3}+\frac{3x^{2}}{2}+\int \sin(x)\mathrm{d}x$$

Natijani olish uchun umumiy integrallar jadvalidagi \int \sin(x)\mathrm{d}x=-\cos(x) integralidan foydalaning.

$$x^{3}+\frac{3x^{2}}{2}-\cos(x)$$

Xos integral bu integral hisoblashning yuqori chegarasida hisoblangan ifodaning boshlangʻich holatidan chiqarib tashlagan holda integral hisoblashning quyi chegarasida hisoblangan ifodaning boshlangʻich holatidir.

$$1^{3}+\frac{3}{2}\times 1^{2}-\cos(1)-\left(0^{3}+\frac{3}{2}\times 0^{2}-\cos(0)\right)$$

Qisqartirish.

$$\frac{1}{2}\left(7-2\cos(1)\right)$$