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

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

I-evaluate

$$\frac{112595}{4}=28148.75$$

Mga Hakbang ng Solution

Pagsamahin ang -2\left(x-2\right) at -\frac{x-2}{2} para makuha ang -\frac{5}{2}\left(x-2\right).

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

Gamitin ang distributive property para i-multiply ang -\frac{5}{2} gamit ang x-2.

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

Ipakita ang -\frac{5}{2}\left(-2\right) bilang isang single fraction.

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

I-multiply ang -5 at -2 para makuha ang 10.

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

I-divide ang 10 gamit ang 2 para makuha ang 5.

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

Pagsamahin ang 4112x at -\frac{5}{2}x para makuha ang \frac{8219}{2}x.

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

Gamitin ang distributive property para i-multiply ang \frac{8219}{2}x+5 gamit ang \frac{7}{23}.

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

I-multiply ang \frac{8219}{2} sa \frac{7}{23} sa pamamagitan ng pag-multiply ng numerator sa numerator at denominator sa denominator.

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

Gawin ang mga multiplication sa fraction na \frac{8219\times 7}{2\times 23}.

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

Ipakita ang 5\times \frac{7}{23} bilang isang single fraction.

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

I-multiply ang 5 at 7 para makuha ang 35.

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

Suriin muna ang indefinite na integral.

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

I-integrate ang sum sa bawat term.

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

I-factor out ang constant sa bawat mga term.

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

Simula \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} para sa k\neq -1, palitan ang \int x\mathrm{d}x ng \frac{x^{2}}{2}. I-multiply ang \frac{57533}{46} times \frac{x^{2}}{2}.

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

Hanapin ang integral ng \frac{35}{23} gamit ang alituntunin ng talahanayan ng mga karaniwang integral \int a\mathrm{d}x=ax.

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

Ang definite integral ay ang antiderivative ng expression na na-evaluate sa upper limit ng integration minus ang antiderivative na na-evaluate sa lower limit ng integration.

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

Pasimplehin.

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