Solve 1 / 2 [ 1 / x + 1 / x ^ 2 - 1 / x ^ 2 ( ∫ frac { 1 } { x } ) ]

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

Evaluate

$$\frac{-\ln(|x|)+x+1-С}{2x^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and x^{2} is x^{2}. Multiply \frac{1}{x} times \frac{x}{x}.

$$\frac{1}{2}\left(\frac{x}{x^{2}}+\frac{1}{x^{2}}-\frac{1}{x^{2}}\int \frac{1}{x}\mathrm{d}x\right)$$

Since \frac{x}{x^{2}} and \frac{1}{x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}\left(\frac{x+1}{x^{2}}-\frac{1}{x^{2}}\int \frac{1}{x}\mathrm{d}x\right)$$

Use the distributive property to multiply \frac{1}{2} by \frac{x+1}{x^{2}}-\frac{1}{x^{2}}\int \frac{1}{x}\mathrm{d}x.

$$\frac{1}{2}\times \frac{x+1}{x^{2}}-\frac{1}{2}\times \frac{1}{x^{2}}\left(1+С\right)$$

Multiply \frac{1}{2} times \frac{x+1}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x+1}{2x^{2}}-\frac{1}{2}\times \frac{1}{x^{2}}\left(1+С\right)$$

Multiply -\frac{1}{2} times \frac{1}{x^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{x+1}{2x^{2}}+\frac{-1}{2x^{2}}\left(1+С\right)$$

Express \frac{-1}{2x^{2}}\left(1+С\right) as a single fraction.

$$\frac{x+1}{2x^{2}}+\frac{-\left(1+С\right)}{2x^{2}}$$

Since \frac{x+1}{2x^{2}} and \frac{-\left(1+С\right)}{2x^{2}} have the same denominator, add them by adding their numerators.

$$\frac{x+1-\left(1+С\right)}{2x^{2}}$$

Do the multiplications in x+1-\left(1+С\right).

$$\frac{x+1-1-С}{2x^{2}}$$

Combine like terms in x+1-1-С.

$$\frac{x-С}{2x^{2}}$$

Differentiate w.r.t. x

$$\frac{\ln(|x|)-\frac{x}{2}-\frac{3}{2}+С}{x^{3}}$$