∫ (from 1 to 8) of (1/sqrt[3]{x}-3x^15) wrt x eching

$$\int _ { 1 } ^ { 8 } ( \frac { 1 } { \sqrt[ 3 ] { x } } - 3 x ^ { 15 } ) d x$$

Baholash

$$-\frac{844424930131893}{16}=-5,277655813 \cdot 10^{13}$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \frac{1}{\sqrt[3]{x}}-3x^{15}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int \frac{1}{\sqrt[3]{x}}\mathrm{d}x+\int -3x^{15}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int \frac{1}{\sqrt[3]{x}}\mathrm{d}x-3\int x^{15}\mathrm{d}x$$

\frac{1}{\sqrt[3]{x}} ni x^{-\frac{1}{3}} sifatida qaytadan yozish. k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{-\frac{1}{3}}\mathrm{d}x integralni \frac{x^{\frac{2}{3}}}{\frac{2}{3}} bilan almashtiring. Qisqartirish.

$$\frac{3x^{\frac{2}{3}}}{2}-3\int x^{15}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x^{15}\mathrm{d}x integralni \frac{x^{16}}{16} bilan almashtiring. -3 ni \frac{x^{16}}{16} marotabaga ko'paytirish.

$$\frac{3x^{\frac{2}{3}}}{2}-\frac{3x^{16}}{16}$$

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

$$\frac{3}{2}\times 8^{\frac{2}{3}}-\frac{3}{16}\times 8^{16}-\left(\frac{3}{2}\times 1^{\frac{2}{3}}-\frac{3}{16}\times 1^{16}\right)$$

Qisqartirish.

$$-\frac{844424930131893}{16}$$