Solve ∫ (from 1 to 3) of (2x-1/x) wrt x

$$\int _ { 1 } ^ { 3 } ( 2 x - \frac { 1 } { x } ) d x$$

Evaluate

$$8-\ln(3)\approx 6.901387711$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2x-\frac{1}{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int -\frac{1}{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x-\int \frac{1}{x}\mathrm{d}x$$

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

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

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$x^{2}-\ln(|x|)$$

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.

$$3^{2}-\ln(|3|)-\left(1^{2}-\ln(|1|)\right)$$

Simplify.

$$8-\ln(3)$$