Issolvi ∫ (from 0 to 3) of (13.5+9.5x)(6+x) wrt x

$$\int_{ 0 }^{ 3 } (13.5+9.5x)(6+x) d x$$

Evalwa

$$645.75$$

Passi tas-Soluzzjonijiet

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 13.5+9.5x b'kull terminu ta' 6+x.

$$\int _{0}^{3}81+13.5x+57x+9.5x^{2}\mathrm{d}x$$

Ikkombina 13.5x u 57x biex tikseb 70.5x.

$$\int _{0}^{3}81+70.5x+9.5x^{2}\mathrm{d}x$$

L-ewwel evalwa l-integru indefinit.

$$\int 81+\frac{141x}{2}+\frac{19x^{2}}{2}\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int 81\mathrm{d}x+\int \frac{141x}{2}\mathrm{d}x+\int \frac{19x^{2}}{2}\mathrm{d}x$$

Iffattura ‘l barra l-kostanti f’kull wieħed minn dawn it-termini.

$$\int 81\mathrm{d}x+\frac{141\int x\mathrm{d}x}{2}+\frac{19\int x^{2}\mathrm{d}x}{2}$$

Sib l-integru ta' 81 billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

$$81x+\frac{141\int x\mathrm{d}x}{2}+\frac{19\int x^{2}\mathrm{d}x}{2}$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x\mathrm{d}x ma' \frac{x^{2}}{2}. Immultiplika 70.5 b'\frac{x^{2}}{2}.

$$81x+\frac{141x^{2}}{4}+\frac{19\int x^{2}\mathrm{d}x}{2}$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{2}\mathrm{d}x ma' \frac{x^{3}}{3}. Immultiplika 9.5 b'\frac{x^{3}}{3}.

$$81x+\frac{141x^{2}}{4}+\frac{19x^{3}}{6}$$

L-integru definit huwa l-antiderivattiv tal-espressjoni evalwata fil-limitu superjuri tal-integrazzjoni minus l-antiderivattiv evalwat fil-limitu inferjuri tal-integrazzjoni.

$$81\times 3+\frac{141}{4}\times 3^{2}+\frac{19}{6}\times 3^{3}-\left(81\times 0+\frac{141}{4}\times 0^{2}+\frac{19}{6}\times 0^{3}\right)$$

Issimplifika.

$$\frac{2583}{4}$$