Datrys ∫ (from 0 to 2) of left(xleft(x-2right)^2right)^2 wrt x

$$\int_{ 0 }^{ 2 } { \left(x { \left(x-2 \right) }^{ 2 } \right) }^{ 2 } d x$$

Enrhifo

$$\frac{128}{105}\approx 1.219047619$$

Camau Ateb

Defnyddio'r theorem binomaidd \left(a-b\right)^{2}=a^{2}-2ab+b^{2} i ehangu'r \left(x-2\right)^{2}.

$$\int _{0}^{2}\left(x\left(x^{2}-4x+4\right)\right)^{2}\mathrm{d}x$$

Defnyddio’r briodwedd ddosbarthu i luosi x â x^{2}-4x+4.

$$\int _{0}^{2}\left(x^{3}-4x^{2}+4x\right)^{2}\mathrm{d}x$$

Sgwâr x^{3}-4x^{2}+4x.

$$\int _{0}^{2}x^{6}-8x^{5}+24x^{4}-32x^{3}+16x^{2}\mathrm{d}x$$

Gwerthuso’r integryn amhenodol yn gyntaf.

$$\int x^{6}-8x^{5}+24x^{4}-32x^{3}+16x^{2}\mathrm{d}x$$

Integreiddio'r swm fesul term.

$$\int x^{6}\mathrm{d}x+\int -8x^{5}\mathrm{d}x+\int 24x^{4}\mathrm{d}x+\int -32x^{3}\mathrm{d}x+\int 16x^{2}\mathrm{d}x$$

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

$$\int x^{6}\mathrm{d}x-8\int x^{5}\mathrm{d}x+24\int x^{4}\mathrm{d}x-32\int x^{3}\mathrm{d}x+16\int x^{2}\mathrm{d}x$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{6}\mathrm{d}x gyda \frac{x^{7}}{7}.

$$\frac{x^{7}}{7}-8\int x^{5}\mathrm{d}x+24\int x^{4}\mathrm{d}x-32\int x^{3}\mathrm{d}x+16\int x^{2}\mathrm{d}x$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{5}\mathrm{d}x gyda \frac{x^{6}}{6}. Lluoswch -8 â \frac{x^{6}}{6}.

$$\frac{x^{7}}{7}-\frac{4x^{6}}{3}+24\int x^{4}\mathrm{d}x-32\int x^{3}\mathrm{d}x+16\int x^{2}\mathrm{d}x$$

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 24 â \frac{x^{5}}{5}.

$$\frac{x^{7}}{7}-\frac{4x^{6}}{3}+\frac{24x^{5}}{5}-32\int x^{3}\mathrm{d}x+16\int x^{2}\mathrm{d}x$$

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x^{3}\mathrm{d}x gyda \frac{x^{4}}{4}. Lluoswch -32 â \frac{x^{4}}{4}.

$$\frac{x^{7}}{7}-\frac{4x^{6}}{3}+\frac{24x^{5}}{5}-8x^{4}+16\int x^{2}\mathrm{d}x$$

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 16 â \frac{x^{3}}{3}.

$$\frac{x^{7}}{7}-\frac{4x^{6}}{3}+\frac{24x^{5}}{5}-8x^{4}+\frac{16x^{3}}{3}$$

Symleiddio.

$$\frac{16x^{3}}{3}-8x^{4}+\frac{24x^{5}}{5}-\frac{4x^{6}}{3}+\frac{x^{7}}{7}$$

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.

$$\frac{16}{3}\times 2^{3}-8\times 2^{4}+\frac{24}{5}\times 2^{5}-\frac{4}{3}\times 2^{6}+\frac{2^{7}}{7}-\left(\frac{16}{3}\times 0^{3}-8\times 0^{4}+\frac{24}{5}\times 0^{5}-\frac{4}{3}\times 0^{6}+\frac{0^{7}}{7}\right)$$

Symleiddio.

$$\frac{128}{105}$$