Solve ∫ (from 273 to 573) of 26.78+42.68*10^-3*x-146.4*10^-7*x^2 wrt x

$$\int_{ 273 }^{ 573 } 26.78+42.68 \times { 10 }^{ -3 } \times x-146.4 \times { 10 }^{ -7 } \times { x }^{ 2 } d x$$

Evaluate

$$12631.295832$$

Solution Steps

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$\int _{273}^{573}26.78+42.68\times \frac{1}{1000}x-146.4\times 10^{-7}x^{2}\mathrm{d}x$$

Multiply 42.68 and \frac{1}{1000} to get \frac{1067}{25000}.

$$\int _{273}^{573}26.78+\frac{1067}{25000}x-146.4\times 10^{-7}x^{2}\mathrm{d}x$$

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$\int _{273}^{573}26.78+\frac{1067}{25000}x-146.4\times \frac{1}{10000000}x^{2}\mathrm{d}x$$

Multiply 146.4 and \frac{1}{10000000} to get \frac{183}{12500000}.

$$\int _{273}^{573}26.78+\frac{1067}{25000}x-\frac{183}{12500000}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 26.78+\frac{1067x}{25000}-\frac{183x^{2}}{12500000}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 26.78\mathrm{d}x+\int \frac{1067x}{25000}\mathrm{d}x+\int -\frac{183x^{2}}{12500000}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 26.78\mathrm{d}x+\frac{1067\int x\mathrm{d}x}{25000}-\frac{183\int x^{2}\mathrm{d}x}{12500000}$$

Find the integral of 26.78 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{1339x}{50}+\frac{1067\int x\mathrm{d}x}{25000}-\frac{183\int x^{2}\mathrm{d}x}{12500000}$$

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{1067}{25000} times \frac{x^{2}}{2}.

$$\frac{1339x}{50}+\frac{1067x^{2}}{50000}-\frac{183\int x^{2}\mathrm{d}x}{12500000}$$

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{183}{12500000} times \frac{x^{3}}{3}.

$$\frac{1339x}{50}+\frac{1067x^{2}}{50000}-\frac{61x^{3}}{12500000}$$

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.

$$26.78\times 573+\frac{1067}{50000}\times 573^{2}-\frac{61}{12500000}\times 573^{3}-\left(26.78\times 273+\frac{1067}{50000}\times 273^{2}-\frac{61}{12500000}\times 273^{3}\right)$$

Simplify.

$$\frac{1578911979}{125000}$$