Issolvi ∫ (-x^2-4x-2)(-3x-7) wrt x

$$\int ( - x ^ { 2 } - 4 x - 2 ) ( - 3 x - 7 ) d x$$

Evalwa

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

Passi tas-Soluzzjonijiet

Uża l-propjetà distributtiva biex timmultiplika -x^{2}-4x-2 b'-3x-7 u kkombina termini simili.

$$\int -3\left(-x^{2}\right)x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

Immultiplika -3 u -1 biex tikseb 3.

$$\int 3x^{2}x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

Biex timmultiplika l-qawwa tal-istess bażi, żid l-esponenti tagħhom. Żid 2 u 1 biex tikseb 3.

$$\int 3x^{3}-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

Immultiplika -7 u -1 biex tikseb 7.

$$\int 3x^{3}+7x^{2}+12x^{2}+34x+14\mathrm{d}x$$

Ikkombina 7x^{2} u 12x^{2} biex tikseb 19x^{2}.

$$\int 3x^{3}+19x^{2}+34x+14\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int 3x^{3}\mathrm{d}x+\int 19x^{2}\mathrm{d}x+\int 34x\mathrm{d}x+\int 14\mathrm{d}x$$

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

$$3\int x^{3}\mathrm{d}x+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

Minn \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} għal k\neq -1, issostitwixxi \int x^{3}\mathrm{d}x ma' \frac{x^{4}}{4}. Immultiplika 3 b'\frac{x^{4}}{4}.

$$\frac{3x^{4}}{4}+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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 19 b'\frac{x^{3}}{3}.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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 34 b'\frac{x^{2}}{2}.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+\int 14\mathrm{d}x$$

Sib l-integru ta' 14 billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x$$

Jekk F\left(x\right) huwa antiderivattiv ta' f\left(x\right), allura s-sett tal-antiderivati kollha ta' f\left(x\right) jingħata minn F\left(x\right)+C. Għalhekk, żid il-kostanti ta' integrazzjoni C\in \mathrm{R} mar-riżultat.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

Iddifferenzja w.r.t. x

$$\left(3x+7\right)\left(x^{2}+4x+2\right)$$