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

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

Evaluate

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

Solution Steps

Combine -2\left(x-2\right) and -\frac{x-2}{2} to get -\frac{5}{2}\left(x-2\right).

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

Use the distributive property to multiply -\frac{5}{2} by x-2.

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

Express -\frac{5}{2}\left(-2\right) as a single fraction.

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

Multiply -5 and -2 to get 10.

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

Divide 10 by 2 to get 5.

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

Combine 4112x and -\frac{5}{2}x to get \frac{8219}{2}x.

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

Use the distributive property to multiply \frac{8219}{2}x+5 by \frac{7}{23}.

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

Multiply \frac{8219}{2} times \frac{7}{23} by multiplying numerator times numerator and denominator times denominator.

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

Do the multiplications in the fraction \frac{8219\times 7}{2\times 23}.

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

Express 5\times \frac{7}{23} as a single fraction.

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

Multiply 5 and 7 to get 35.

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

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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{57533}{46} times \frac{x^{2}}{2}.

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

Find the integral of \frac{35}{23} using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

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

Simplify.

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