Issolvi ∫ (from - 0.15 to 665) of [(-x^2+2x)-(-1+1/2x)] wrt x

$$\int _ { - 0.15 } ^ { 665 } [ ( - x ^ { 2 } + 2 x ) - ( - 1 + \frac { 1 } { 2 } x ) ] d x$$

Evalwa

$$-\frac{146541311677}{1500}\approx -97694207.784666667$$

Passi tas-Soluzzjonijiet

Biex issib l-oppost ta' -1+\frac{1}{2}x, sib l-oppost ta' kull terminu.

$$\int _{-0.15}^{665}-x^{2}+2x+1-\frac{1}{2}x\mathrm{d}x$$

Ikkombina 2x u -\frac{1}{2}x biex tikseb \frac{3}{2}x.

$$\int _{-0.15}^{665}-x^{2}+\frac{3}{2}x+1\mathrm{d}x$$

L-ewwel evalwa l-integru indefinit.

$$\int -x^{2}+\frac{3x}{2}+1\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int -x^{2}\mathrm{d}x+\int \frac{3x}{2}\mathrm{d}x+\int 1\mathrm{d}x$$

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

$$-\int x^{2}\mathrm{d}x+\frac{3\int x\mathrm{d}x}{2}+\int 1\mathrm{d}x$$

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 -1 b'\frac{x^{3}}{3}.

$$-\frac{x^{3}}{3}+\frac{3\int x\mathrm{d}x}{2}+\int 1\mathrm{d}x$$

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 \frac{3}{2} b'\frac{x^{2}}{2}.

$$-\frac{x^{3}}{3}+\frac{3x^{2}}{4}+\int 1\mathrm{d}x$$

Sib l-integru ta' 1 billi tuża t-tabella tar-regola tal-integrali komuni \int a\mathrm{d}x=ax.

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

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

$$-\frac{665^{3}}{3}+\frac{3}{4}\times 665^{2}+665-\left(-\frac{\left(-0.15\right)^{3}}{3}+\frac{3}{4}\left(-0.15\right)^{2}-0.15\right)$$

Issimplifika.

$$-\frac{146541311677}{1500}$$