Solve ∫ (from 0 to 0.00375) of 500-4/3*10^5x wrt x

$$\int_{ 0 }^{ 0.00375 } 500- \frac{ 4 }{ 3 } \times { 10 }^{ 5 } x d x$$

Evaluate

$$0.9375$$

Solution Steps

Calculate 10 to the power of 5 and get 100000.

$$\int _{0}^{0.00375}500-\frac{4}{3}\times 100000x\mathrm{d}x$$

Express \frac{4}{3}\times 100000 as a single fraction.

$$\int _{0}^{0.00375}500-\frac{4\times 100000}{3}x\mathrm{d}x$$

Multiply 4 and 100000 to get 400000.

$$\int _{0}^{0.00375}500-\frac{400000}{3}x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 500-\frac{400000x}{3}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 500\mathrm{d}x+\int -\frac{400000x}{3}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 500\mathrm{d}x-\frac{400000\int x\mathrm{d}x}{3}$$

Find the integral of 500 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$500x-\frac{400000\int x\mathrm{d}x}{3}$$

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{400000}{3} times \frac{x^{2}}{2}.

$$500x-\frac{200000x^{2}}{3}$$

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.

$$500\times 0.00375-\frac{200000}{3}\times 0.00375^{2}-\left(500\times 0-\frac{200000}{3}\times 0^{2}\right)$$

Simplify.

$$\frac{15}{16}$$