Solve ∫ (from 373.14 to 453.14) of (29.16+14.49div1000x-2.022div1000000*x^2)*1000div18 wrt x

$$\int_{ 373.14 }^{ 453.14 } (29.16+14.49 \div 1000x-2.022 \div 1000000 \times { x }^{ 2 } ) \times 1000 \div 18 d x$$

Evaluate

$$\frac{43500244772861}{281250000}\approx 154667.536970172$$

Solution Steps

Expand \frac{14.49}{1000} by multiplying both numerator and the denominator by 100.

$$\int _{373.14}^{453.14}\frac{\left(29.16+\frac{1449}{100000}x-\frac{2.022}{1000000}x^{2}\right)\times 1000}{18}\mathrm{d}x$$

Expand \frac{2.022}{1000000} by multiplying both numerator and the denominator by 1000.

$$\int _{373.14}^{453.14}\frac{\left(29.16+\frac{1449}{100000}x-\frac{2022}{1000000000}x^{2}\right)\times 1000}{18}\mathrm{d}x$$

Reduce the fraction \frac{2022}{1000000000} to lowest terms by extracting and canceling out 2.

$$\int _{373.14}^{453.14}\frac{\left(29.16+\frac{1449}{100000}x-\frac{1011}{500000000}x^{2}\right)\times 1000}{18}\mathrm{d}x$$

Divide \left(29.16+\frac{1449}{100000}x-\frac{1011}{500000000}x^{2}\right)\times 1000 by 18 to get \left(29.16+\frac{1449}{100000}x-\frac{1011}{500000000}x^{2}\right)\times \frac{500}{9}.

$$\int _{373.14}^{453.14}\left(29.16+\frac{1449}{100000}x-\frac{1011}{500000000}x^{2}\right)\times \frac{500}{9}\mathrm{d}x$$

Use the distributive property to multiply 29.16+\frac{1449}{100000}x-\frac{1011}{500000000}x^{2} by \frac{500}{9}.

$$\int _{373.14}^{453.14}1620+\frac{161}{200}x-\frac{337}{3000000}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 1620+\frac{161x}{200}-\frac{337x^{2}}{3000000}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1620\mathrm{d}x+\int \frac{161x}{200}\mathrm{d}x+\int -\frac{337x^{2}}{3000000}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 1620\mathrm{d}x+\frac{161\int x\mathrm{d}x}{200}-\frac{337\int x^{2}\mathrm{d}x}{3000000}$$

Find the integral of 1620 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$1620x+\frac{161\int x\mathrm{d}x}{200}-\frac{337\int x^{2}\mathrm{d}x}{3000000}$$

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{161}{200} times \frac{x^{2}}{2}.

$$1620x+\frac{161x^{2}}{400}-\frac{337\int x^{2}\mathrm{d}x}{3000000}$$

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{337}{3000000} times \frac{x^{3}}{3}.

$$1620x+\frac{161x^{2}}{400}-\frac{337x^{3}}{9000000}$$

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.

$$1620\times 453.14+\frac{161}{400}\times 453.14^{2}-\frac{337}{9000000}\times 453.14^{3}-\left(1620\times 373.14+\frac{161}{400}\times 373.14^{2}-\frac{337}{9000000}\times 373.14^{3}\right)$$

Simplify.

$$\frac{43500244772861}{281250000}$$