Solve ∫ (from 2 to 7) of (41.12x-(2(x-2))(-x-2/2))(7/2.3)

$$\int _ { 2 } ^ { 7 } ( 41.12 x - ( 2 ( x - 2 ) ) ( - \frac { x - 2 } { 2 } ) ) ( \frac { 7 } { 2.3 } )$$

Evaluate

$$\frac{203042}{69}\approx 2942.637681159$$

Solution Steps

Cancel out 2 and 2.

$$\int _{2}^{7}\left(41.12x-\left(-\left(x-2\right)\left(x-2\right)\right)\right)\times \frac{7}{2.3}\mathrm{d}x$$

Use the distributive property to multiply -\left(x-2\right) by x-2.

$$\int _{2}^{7}\left(41.12x-\left(-\left(x-2\right)x+2x-4\right)\right)\times \frac{7}{2.3}\mathrm{d}x$$

Use the distributive property to multiply -1 by x-2.

$$\int _{2}^{7}\left(41.12x-\left(\left(-x+2\right)x+2x-4\right)\right)\times \frac{7}{2.3}\mathrm{d}x$$

Use the distributive property to multiply -x+2 by x.

$$\int _{2}^{7}\left(41.12x-\left(-x^{2}+2x+2x-4\right)\right)\times \frac{7}{2.3}\mathrm{d}x$$

Combine 2x and 2x to get 4x.

$$\int _{2}^{7}\left(41.12x-\left(-x^{2}+4x-4\right)\right)\times \frac{7}{2.3}\mathrm{d}x$$

To find the opposite of -x^{2}+4x-4, find the opposite of each term.

$$\int _{2}^{7}\left(41.12x-\left(-x^{2}\right)-4x-\left(-4\right)\right)\times \frac{7}{2.3}\mathrm{d}x$$

The opposite of -x^{2} is x^{2}.

$$\int _{2}^{7}\left(41.12x+x^{2}-4x-\left(-4\right)\right)\times \frac{7}{2.3}\mathrm{d}x$$

The opposite of -4 is 4.

$$\int _{2}^{7}\left(41.12x+x^{2}-4x+4\right)\times \frac{7}{2.3}\mathrm{d}x$$

Combine 41.12x and -4x to get 37.12x.

$$\int _{2}^{7}\left(37.12x+x^{2}+4\right)\times \frac{7}{2.3}\mathrm{d}x$$

Expand \frac{7}{2.3} by multiplying both numerator and the denominator by 10.

$$\int _{2}^{7}\left(37.12x+x^{2}+4\right)\times \frac{70}{23}\mathrm{d}x$$

Use the distributive property to multiply 37.12x+x^{2}+4 by \frac{70}{23}.

$$\int _{2}^{7}37.12x\times \frac{70}{23}+x^{2}\times \frac{70}{23}+4\times \frac{70}{23}\mathrm{d}x$$

Convert decimal number 37.12 to fraction \frac{3712}{100}. Reduce the fraction \frac{3712}{100} to lowest terms by extracting and canceling out 4.

$$\int _{2}^{7}\frac{928}{25}x\times \frac{70}{23}+x^{2}\times \frac{70}{23}+4\times \frac{70}{23}\mathrm{d}x$$

Multiply \frac{928}{25} times \frac{70}{23} by multiplying numerator times numerator and denominator times denominator.

$$\int _{2}^{7}\frac{928\times 70}{25\times 23}x+x^{2}\times \frac{70}{23}+4\times \frac{70}{23}\mathrm{d}x$$

Do the multiplications in the fraction \frac{928\times 70}{25\times 23}.

$$\int _{2}^{7}\frac{64960}{575}x+x^{2}\times \frac{70}{23}+4\times \frac{70}{23}\mathrm{d}x$$

Reduce the fraction \frac{64960}{575} to lowest terms by extracting and canceling out 5.

$$\int _{2}^{7}\frac{12992}{115}x+x^{2}\times \frac{70}{23}+4\times \frac{70}{23}\mathrm{d}x$$

Express 4\times \frac{70}{23} as a single fraction.

$$\int _{2}^{7}\frac{12992}{115}x+x^{2}\times \frac{70}{23}+\frac{4\times 70}{23}\mathrm{d}x$$

Multiply 4 and 70 to get 280.

$$\int _{2}^{7}\frac{12992}{115}x+x^{2}\times \frac{70}{23}+\frac{280}{23}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int \frac{12992x}{115}+\frac{70x^{2}}{23}+\frac{280}{23}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \frac{12992x}{115}\mathrm{d}x+\int \frac{70x^{2}}{23}\mathrm{d}x+\int \frac{280}{23}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\frac{12992\int x\mathrm{d}x}{115}+\frac{70\int x^{2}\mathrm{d}x}{23}+\int \frac{280}{23}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}. Multiply \frac{12992}{115} times \frac{x^{2}}{2}.

$$\frac{6496x^{2}}{115}+\frac{70\int x^{2}\mathrm{d}x}{23}+\int \frac{280}{23}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply \frac{70}{23} times \frac{x^{3}}{3}.

$$\frac{6496x^{2}}{115}+\frac{70x^{3}}{69}+\int \frac{280}{23}\mathrm{d}x$$

Find the integral of \frac{280}{23} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{6496x^{2}}{115}+\frac{70x^{3}}{69}+\frac{280x}{23}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\frac{6496}{115}\times 7^{2}+\frac{70}{69}\times 7^{3}+\frac{280}{23}\times 7-\left(\frac{6496}{115}\times 2^{2}+\frac{70}{69}\times 2^{3}+\frac{280}{23}\times 2\right)$$

Simplify.

$$\frac{203042}{69}$$