Solve ∫ (from 300 to 400) of (26+33*10^-2*x-3*1*10^-5*x^2) wrt x

$$\int _ { 300 } ^ { 400 } ( 26 + 33 \times 10 ^ { - 2 } \cdot x - 3 \cdot 1 \times 10 ^ { - 5 } \cdot x ^ { 2 } ) d x$$

Evaluate

$$13780$$

Solution Steps

Calculate 10 to the power of -2 and get \frac{1}{100}.

$$\int _{300}^{400}26+33\times \frac{1}{100}x-3\times 1\times 10^{-5}x^{2}\mathrm{d}x$$

Multiply 33 and \frac{1}{100} to get \frac{33}{100}.

$$\int _{300}^{400}26+\frac{33}{100}x-3\times 1\times 10^{-5}x^{2}\mathrm{d}x$$

Multiply 3 and 1 to get 3.

$$\int _{300}^{400}26+\frac{33}{100}x-3\times 10^{-5}x^{2}\mathrm{d}x$$

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$\int _{300}^{400}26+\frac{33}{100}x-3\times \frac{1}{100000}x^{2}\mathrm{d}x$$

Multiply 3 and \frac{1}{100000} to get \frac{3}{100000}.

$$\int _{300}^{400}26+\frac{33}{100}x-\frac{3}{100000}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 26+\frac{33x}{100}-\frac{3x^{2}}{100000}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 26\mathrm{d}x+\int \frac{33x}{100}\mathrm{d}x+\int -\frac{3x^{2}}{100000}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 26\mathrm{d}x+\frac{33\int x\mathrm{d}x}{100}-\frac{3\int x^{2}\mathrm{d}x}{100000}$$

Find the integral of 26 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$26x+\frac{33\int x\mathrm{d}x}{100}-\frac{3\int x^{2}\mathrm{d}x}{100000}$$

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{33}{100} times \frac{x^{2}}{2}.

$$26x+\frac{33x^{2}}{200}-\frac{3\int x^{2}\mathrm{d}x}{100000}$$

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{3}{100000} times \frac{x^{3}}{3}.

$$26x+\frac{33x^{2}}{200}-\frac{x^{3}}{100000}$$

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.

$$26\times 400+\frac{33}{200}\times 400^{2}-\frac{400^{3}}{100000}-\left(26\times 300+\frac{33}{200}\times 300^{2}-\frac{300^{3}}{100000}\right)$$

Simplify.

$$13780$$