Solve ∫ (from 1 to 2) of y-1/y wrt y

$$\int _ { 1 } ^ { 2 } y - \frac { 1 } { y } d y$$

Evaluate

$$\frac{3}{2}-\ln(2)\approx 0.806852819$$

Solution Steps

Evaluate the indefinite integral first.

$$\int y-\frac{1}{y}\mathrm{d}y$$

Integrate the sum term by term.

$$\int y\mathrm{d}y+\int -\frac{1}{y}\mathrm{d}y$$

Factor out the constant in each of the terms.

$$\int y\mathrm{d}y-\int \frac{1}{y}\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y\mathrm{d}y with \frac{y^{2}}{2}.

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

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

$$\frac{y^{2}}{2}-\ln(|y|)$$

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.

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

Simplify.

$$\frac{3}{2}-\ln(2)$$