Solve ∫ (from 1 to 2) of 2x-6 wrt x

$$\int _ { 1 } ^ { 2 } 2 x - 6 d x$$

Evaluate

$$-3$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2x-6\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int -6\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x+\int -6\mathrm{d}x$$

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

$$x^{2}+\int -6\mathrm{d}x$$

Find the integral of -6 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{2}-6x$$

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.

$$2^{2}-6\times 2-\left(1^{2}-6\right)$$

Simplify.

$$-3$$