∫ (from -3 to frac{ 1) of {2}}-2x^2-5x+3 wrt x eching

$$\int_{ -3 }^{ \frac{ 1 }{ 2 } } -2 { x }^{ 2 } -5x+3 d x$$

Baholash

$$\frac{343}{24}\approx 14,291666667$$

Yechim qadamlari

Avval noaniq integralni baholang.

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

Summani muddatma-muddat integratsiya qiling.

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

Har bir shartda konstantani qavsdan tashqariga oling.

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

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

$$-\frac{2x^{3}}{3}-5\int x\mathrm{d}x+\int 3\mathrm{d}x$$

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. -5 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$-\frac{2x^{3}}{3}-\frac{5x^{2}}{2}+\int 3\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 3 integralini toping.

$$-\frac{2x^{3}}{3}-\frac{5x^{2}}{2}+3x$$

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

$$-\frac{2}{3}\times \left(\frac{1}{2}\right)^{3}-\frac{5}{2}\times \left(\frac{1}{2}\right)^{2}+3\times \frac{1}{2}-\left(-\frac{2}{3}\left(-3\right)^{3}-\frac{5}{2}\left(-3\right)^{2}+3\left(-3\right)\right)$$

Qisqartirish.

$$\frac{343}{24}$$