Datrys ∫ (from 0 to 3) of xleft(x-2right)left(x+3right) wrt x

$$\int_{ 0 }^{ 3 } x \left( x-2 \right) \left( x+3 \right) d x$$

Enrhifo

$$\frac{9}{4}=2.25$$

Camau Ateb

Defnyddio’r briodwedd ddosbarthu i luosi x â x-2.

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

Cyfrifwch y briodoledd ddosrannol drwy luosi pob x^{2}-2x gan bob x+3.

$$\int _{0}^{3}x^{3}+3x^{2}-2x^{2}-6x\mathrm{d}x$$

Cyfuno 3x^{2} a -2x^{2} i gael x^{2}.

$$\int _{0}^{3}x^{3}+x^{2}-6x\mathrm{d}x$$

Gwerthuso’r integryn amhenodol yn gyntaf.

$$\int x^{3}+x^{2}-6x\mathrm{d}x$$

Integreiddio'r swm fesul term.

$$\int x^{3}\mathrm{d}x+\int x^{2}\mathrm{d}x+\int -6x\mathrm{d}x$$

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

$$\int x^{3}\mathrm{d}x+\int x^{2}\mathrm{d}x-6\int x\mathrm{d}x$$

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}.

$$\frac{x^{4}}{4}+\int x^{2}\mathrm{d}x-6\int x\mathrm{d}x$$

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}.

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

Ers \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} ar gyfer k\neq -1, disodli \int x\mathrm{d}x gyda \frac{x^{2}}{2}. Lluoswch -6 â \frac{x^{2}}{2}.

$$\frac{x^{4}}{4}+\frac{x^{3}}{3}-3x^{2}$$

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.

$$\frac{3^{4}}{4}+\frac{3^{3}}{3}-3\times 3^{2}-\left(\frac{0^{4}}{4}+\frac{0^{3}}{3}-3\times 0^{2}\right)$$

Symleiddio.

$$\frac{9}{4}$$