Solve ∫ (from 0 to 2) of 1/2{left(4-1/2xright)}^ wrt x

$$\int_{ 0 }^{ 2 } \frac{ 1 }{ 2 } { \left(4- \frac{ 1 }{ 2 } x \right) }^{ } d x$$

Evaluate

$$\frac{7}{2}=3.5$$

Solution Steps

Calculate 4-\frac{1}{2}x to the power of 1 and get 4-\frac{1}{2}x.

$$\int _{0}^{2}\frac{1}{2}\left(4-\frac{1}{2}x\right)\mathrm{d}x$$

Use the distributive property to multiply \frac{1}{2} by 4-\frac{1}{2}x.

$$\int _{0}^{2}2-\frac{1}{4}x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 2-\frac{x}{4}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2\mathrm{d}x+\int -\frac{x}{4}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 2\mathrm{d}x-\frac{\int x\mathrm{d}x}{4}$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2x-\frac{\int x\mathrm{d}x}{4}$$

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{1}{4} times \frac{x^{2}}{2}.

$$2x-\frac{x^{2}}{8}$$

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.

$$2\times 2-\frac{2^{2}}{8}-\left(2\times 0-\frac{0^{2}}{8}\right)$$

Simplify.

$$\frac{7}{2}$$