Solve ∫ (from 0 to 1) of (0.05-x)/ wrt x

$$\int_{ 0 }^{ 1 } (0.05-x) \frac{ }{ } d x$$

Evaluate

$$-0.45$$

Solution Steps

Divide 1 by 1 to get 1.

$$\int _{0}^{1}\left(0.05-x\right)\times 1\mathrm{d}x$$

Use the distributive property to multiply 0.05-x by 1.

$$\int _{0}^{1}0.05-x\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 0.05-x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 0.05\mathrm{d}x+\int -x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 0.05\mathrm{d}x-\int x\mathrm{d}x$$

Find the integral of 0.05 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x}{20}-\int x\mathrm{d}x$$

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

$$\frac{x}{20}-\frac{x^{2}}{2}$$

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.

$$0.05\times 1-\frac{1^{2}}{2}-\left(0.05\times 0-\frac{0^{2}}{2}\right)$$

Simplify.

$$-\frac{9}{20}$$