Solve ∫ (from 1 / 4 to 3 / 4) of [x]d[2x]

$$\int _ { 1 / 4 } ^ { 3 / 4 } [ x ] d [ 2 x ]$$

Evaluate

$$\frac{13d}{48}$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int _{\frac{1}{4}}^{\frac{3}{4}}x^{2}d\times 2\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 2x^{2}d\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$2d\int x^{2}\mathrm{d}x$$

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}.

$$2d\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{2dx^{3}}{3}$$

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.

$$\frac{2}{3}d\times \left(\frac{3}{4}\right)^{3}-\frac{2}{3}d\times \left(\frac{1}{4}\right)^{3}$$

Simplify.

$$\frac{13d}{48}$$

Differentiate w.r.t. d

$$\frac{13}{48} = 0.2708333333333333$$