Solve ∫ _{1/2}^2(4x-1/x) wrt x

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

Evaluate

$$\frac{15}{2}-2\ln(2)\approx 6.113705639$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x-\frac{1}{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x\mathrm{d}x+\int -\frac{1}{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x\mathrm{d}x-\int \frac{1}{x}\mathrm{d}x$$

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

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

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

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

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.

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

Simplify.

$$\frac{15}{2}-2\ln(2)$$