Datrys ∫ _{frac{sqrt{2}}{2}}^1(x^4-1/2x^2) wrt x

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

Enrhifo

$$\frac{\sqrt{2}}{60}+\frac{1}{30}\approx 0.056903559$$

Camau Ateb

Gwerthuso’r integryn amhenodol yn gyntaf.

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

Integreiddio'r swm fesul term.

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

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

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

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}.

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

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}.

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

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.

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

Symleiddio.

$$\frac{1}{30}+\frac{\sqrt{2}}{60}$$