Issolvi ∫ (from 0 to 3) of (91x+198)(-6-x) wrt x

$$\int_{ 0 }^{ 3 } (91x+198)(-6-x) d x$$

Evalwa

$$-7731$$

Passi tas-Soluzzjonijiet

Applika l-propjetà distributtiva billi timmultiplika kull terminu ta' 91x+198 b'kull terminu ta' -6-x.

$$\int _{0}^{3}-546x-91x^{2}-1188-198x\mathrm{d}x$$

Ikkombina -546x u -198x biex tikseb -744x.

$$\int _{0}^{3}-744x-91x^{2}-1188\mathrm{d}x$$

L-ewwel evalwa l-integru indefinit.

$$\int -744x-91x^{2}-1188\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int -744x\mathrm{d}x+\int -91x^{2}\mathrm{d}x+\int -1188\mathrm{d}x$$

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

$$-744\int x\mathrm{d}x-91\int x^{2}\mathrm{d}x+\int -1188\mathrm{d}x$$

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 -744 b'\frac{x^{2}}{2}.

$$-372x^{2}-91\int x^{2}\mathrm{d}x+\int -1188\mathrm{d}x$$

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 -91 b'\frac{x^{3}}{3}.

$$-372x^{2}-\frac{91x^{3}}{3}+\int -1188\mathrm{d}x$$

Sib l-integru ta' -1188 billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

$$-372x^{2}-\frac{91x^{3}}{3}-1188x$$

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

$$-372\times 3^{2}-\frac{91}{3}\times 3^{3}-1188\times 3-\left(-372\times 0^{2}-\frac{91}{3}\times 0^{3}-1188\times 0\right)$$

Issimplifika.

$$-7731$$