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

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

Luacháil

$$-\frac{844424930131893}{16}=-5.277655813 \cdot 10^{13}$$

Céimeanna Réitigh

Déan luacháil ar an suimeálaí éiginnte ar dtús.

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

Measc an tsuim téarma fá téarma.

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

Fág an leanúnach sna téarmaí as an áireamh.

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

Athscríobh \frac{1}{\sqrt[3]{x}} mar x^{-\frac{1}{3}}. Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x^{-\frac{1}{3}}\mathrm{d}x le \frac{x^{\frac{2}{3}}}{\frac{2}{3}}. Simpligh.

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

Ó \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} fá choinne k\neq -1, athchuir \int x^{15}\mathrm{d}x le \frac{x^{16}}{16}. Méadaigh -3 faoi \frac{x^{16}}{16}.

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

Is ionann suimeálaí cinnte agus frithdhíorthach an nath luacháilte ag teorainn uachtair na suimeála lúide an frithdhíorthach luacháilte ag teorainn íochtair na suimeála.

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

Simpligh.

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