Solve ∫ (1/x+1/x^2)

$$\int ( \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } )$$

Evaluate

$$\ln(|x|)-\frac{1}{x}+С$$

Solution Steps

Integrate the sum term by term.

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

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

$$\ln(|x|)+\int \frac{1}{x^{2}}\mathrm{d}x$$

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

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

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

$$\ln(|x|)-\frac{1}{x}+С$$

Differentiate w.r.t. x

$$\frac{x+1}{x^{2}}$$