Rešite ∫ (from 2 to 7) of left(4112x-2left(x-2right)-x-2/2right)7/23 wrt x

$$\int_{ 2 }^{ 7 } \left( 4112x-2 \left( x-2 \right) - \frac{ x-2 }{ 2 } \right) \frac{ 7 }{ 23 } d x$$

Ovrednoti

$$\frac{112595}{4}=28148,75$$

Koraki rešitve

Združite -2\left(x-2\right) in -\frac{x-2}{2}, da dobite -\frac{5}{2}\left(x-2\right).

$$\int _{2}^{7}\left(4112x-\frac{5}{2}\left(x-2\right)\right)\times \frac{7}{23}\mathrm{d}x$$

Uporabite distributivnost, da pomnožite -\frac{5}{2} s/z x-2.

$$\int _{2}^{7}\left(4112x-\frac{5}{2}x-\frac{5}{2}\left(-2\right)\right)\times \frac{7}{23}\mathrm{d}x$$

Izrazite -\frac{5}{2}\left(-2\right) kot enojni ulomek.

$$\int _{2}^{7}\left(4112x-\frac{5}{2}x+\frac{-5\left(-2\right)}{2}\right)\times \frac{7}{23}\mathrm{d}x$$

Pomnožite -5 in -2, da dobite 10.

$$\int _{2}^{7}\left(4112x-\frac{5}{2}x+\frac{10}{2}\right)\times \frac{7}{23}\mathrm{d}x$$

Delite 10 s/z 2, da dobite 5.

$$\int _{2}^{7}\left(4112x-\frac{5}{2}x+5\right)\times \frac{7}{23}\mathrm{d}x$$

Združite 4112x in -\frac{5}{2}x, da dobite \frac{8219}{2}x.

$$\int _{2}^{7}\left(\frac{8219}{2}x+5\right)\times \frac{7}{23}\mathrm{d}x$$

Uporabite distributivnost, da pomnožite \frac{8219}{2}x+5 s/z \frac{7}{23}.

$$\int _{2}^{7}\frac{8219}{2}x\times \frac{7}{23}+5\times \frac{7}{23}\mathrm{d}x$$

Pomnožite \frac{8219}{2} s/z \frac{7}{23} tako, da pomnožite števec s števcem in imenovalec z imenovalcem.

$$\int _{2}^{7}\frac{8219\times 7}{2\times 23}x+5\times \frac{7}{23}\mathrm{d}x$$

Izvedite množenja v ulomku \frac{8219\times 7}{2\times 23}.

$$\int _{2}^{7}\frac{57533}{46}x+5\times \frac{7}{23}\mathrm{d}x$$

Izrazite 5\times \frac{7}{23} kot enojni ulomek.

$$\int _{2}^{7}\frac{57533}{46}x+\frac{5\times 7}{23}\mathrm{d}x$$

Pomnožite 5 in 7, da dobite 35.

$$\int _{2}^{7}\frac{57533}{46}x+\frac{35}{23}\mathrm{d}x$$

Najprej ovrednotite nedoločni integral.

$$\int \frac{57533x}{46}+\frac{35}{23}\mathrm{d}x$$

Vsoto povežite z izrazom.

$$\int \frac{57533x}{46}\mathrm{d}x+\int \frac{35}{23}\mathrm{d}x$$

Faktorizirajte konstanto v vseh izrazih.

$$\frac{57533\int x\mathrm{d}x}{46}+\int \frac{35}{23}\mathrm{d}x$$

Ker \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} za k\neq -1, zamenjajte \int x\mathrm{d}x s \frac{x^{2}}{2}. Pomnožite \frac{57533}{46} s/z \frac{x^{2}}{2}.

$$\frac{57533x^{2}}{92}+\int \frac{35}{23}\mathrm{d}x$$

Poiščite integral \frac{35}{23} s tabelo skupnega integrali pravila \int a\mathrm{d}x=ax.

$$\frac{57533x^{2}}{92}+\frac{35x}{23}$$

Določen integral je integral izraza, ovrednotenega pri zgornji omejitvi integriranja, minus integral, ovrednoten pri spodnji omejitvi integriranja.

$$\frac{57533}{92}\times 7^{2}+\frac{35}{23}\times 7-\left(\frac{57533}{92}\times 2^{2}+\frac{35}{23}\times 2\right)$$

Poenostavite.

$$\frac{112595}{4}$$