∫ (from - 15 to 5) of -3x^2+11x+25 wrt x eching

$$\int _ { - 15 } ^ { 5 } - 3 x ^ { 2 } + 11 x + 25 d x$$

Baholash

$$-4100$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int -3x^{2}+11x+25\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int -3x^{2}\mathrm{d}x+\int 11x\mathrm{d}x+\int 25\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$-3\int x^{2}\mathrm{d}x+11\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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.

$$-x^{3}+11\int x\mathrm{d}x+\int 25\mathrm{d}x$$

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

$$-x^{3}+\frac{11x^{2}}{2}+\int 25\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, 25 integralini toping.

$$-x^{3}+\frac{11x^{2}}{2}+25x$$

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

$$-5^{3}+\frac{11}{2}\times 5^{2}+25\times 5-\left(-\left(-15\right)^{3}+\frac{11}{2}\left(-15\right)^{2}+25\left(-15\right)\right)$$

Qisqartirish.

$$-4100$$