Issolvi ∫ (from 0 to 20) of -0.05x+0.05div10*x^2 wrt x

$$\int_{ 0 }^{ 20 } -0.05x+0.05 \div 10 \times { x }^{ 2 } d x$$

Evalwa

$$\frac{10}{3}\approx 3.333333333$$

Passi tas-Soluzzjonijiet

Espandi \frac{0.05}{10} billi timmultiplika kemm in-numeratur kif ukoll id-denominatur b'100.

$$\int _{0}^{20}-0.05x+\frac{5}{1000}x^{2}\mathrm{d}x$$

Naqqas il-frazzjoni \frac{5}{1000} għat-termini l-aktar baxxi billi testratta u tikkanċella barra 5.

$$\int _{0}^{20}-0.05x+\frac{1}{200}x^{2}\mathrm{d}x$$

L-ewwel evalwa l-integru indefinit.

$$\int -\frac{x}{20}+\frac{x^{2}}{200}\mathrm{d}x$$

Tintegra s-somma terminu b'terminu.

$$\int -\frac{x}{20}\mathrm{d}x+\int \frac{x^{2}}{200}\mathrm{d}x$$

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

$$-\frac{\int x\mathrm{d}x}{20}+\frac{\int x^{2}\mathrm{d}x}{200}$$

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 -0.05 b'\frac{x^{2}}{2}.

$$-\frac{x^{2}}{40}+\frac{\int x^{2}\mathrm{d}x}{200}$$

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 \frac{1}{200} b'\frac{x^{3}}{3}.

$$-\frac{x^{2}}{40}+\frac{x^{3}}{600}$$

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

$$-\frac{20^{2}}{40}+\frac{20^{3}}{600}-\left(-\frac{0^{2}}{40}+\frac{0^{3}}{600}\right)$$

Issimplifika.

$$\frac{10}{3}$$