Datrys ∫ (from 0 to sqrt { 2) of / 2 } (1/2x^2-x^4) wrt x

$$\int _ { 0 } ^ { \frac { \sqrt { 2 } } { 2 } } ( \frac { 1 } { 2 } x ^ { 2 } - x ^ { 4 } ) d x$$

Enrhifo

$$\frac{\sqrt{2}}{60}\approx 0.023570226$$

Camau Ateb

Gwerthuso’r integryn amhenodol yn gyntaf.

$$\int \frac{x^{2}}{2}-x^{4}\mathrm{d}x$$

Integreiddio'r swm fesul term.

$$\int \frac{x^{2}}{2}\mathrm{d}x+\int -x^{4}\mathrm{d}x$$

Ffactoreiddio allan y cysonyn ym mhob un o'r termau.

$$\frac{\int x^{2}\mathrm{d}x}{2}-\int x^{4}\mathrm{d}x$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{2}\mathrm{d}x gyda \frac{x^{3}}{3}. Lluoswch \frac{1}{2} â \frac{x^{3}}{3}.

$$\frac{x^{3}}{6}-\int x^{4}\mathrm{d}x$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{4}\mathrm{d}x gyda \frac{x^{5}}{5}. Lluoswch -1 â \frac{x^{5}}{5}.

$$\frac{x^{3}}{6}-\frac{x^{5}}{5}$$

Yr integryn pendant yw integryn amhendant y mynegiant wedi’i werthuso ar lefel uchaf yr integreiddiad llai’r integryn amhendant ar lefel isaf yr integreiddiad.

$$\frac{1}{6}\times \left(\frac{1}{2}\times 2^{\frac{1}{2}}\right)^{3}-\frac{1}{5}\times \left(\frac{1}{2}\times 2^{\frac{1}{2}}\right)^{5}-\left(\frac{0^{3}}{6}-\frac{0^{5}}{5}\right)$$

Symleiddio.

$$\frac{\sqrt{2}}{60}$$