Solve ∫ 1/y^2-1/y wrt y

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

Evaluate

$$-\ln(|y|)-\frac{1}{y}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

$$-\frac{1}{y}-\ln(|y|)$$

If F\left(y\right) is an antiderivative of f\left(y\right), then the set of all antiderivatives of f\left(y\right) is given by F\left(y\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$-\frac{1}{y}-\ln(|y|)+С$$

Differentiate w.r.t. y

$$\frac{1-y}{y^{2}}$$