Solve ∫ 5x-x+2x/x^2 wrt x

$$\int \frac { 5 x - x + 2 x } { x ^ { 2 } } d x$$

Evaluate

$$6\ln(|x|)+С$$

Solution Steps

Combine 5x and -x to get 4x.

$$\int \frac{4x+2x}{x^{2}}\mathrm{d}x$$

Combine 4x and 2x to get 6x.

$$\int \frac{6x}{x^{2}}\mathrm{d}x$$

Cancel out x in both numerator and denominator.

$$\int \frac{6}{x}\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$6\int \frac{1}{x}\mathrm{d}x$$

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

$$6\ln(|x|)$$

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.

$$6\ln(|x|)+С$$

Differentiate w.r.t. x

$$\frac{6}{x}$$