Solve 1/x^2+3x+2+1/x^2-1

$$\frac { 1 } { x ^ { 2 } + 3 x + 2 } + \frac { 1 } { x ^ { 2 } - 1 }$$

Evaluate

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

Short Solution Steps

Factor x^{2}+3x+2. Factor x^{2}-1.

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

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

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

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

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

Combine like terms in x-1+x+2.

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

Expand \left(x-1\right)\left(x+1\right)\left(x+2\right).

$$\frac{2x+1}{x^{3}+2x^{2}-x-2}$$

Differentiate w.r.t. x

$$-\frac{4x^{3}+7x^{2}+4x+3}{\left(\left(x+2\right)\left(x^{2}-1\right)\right)^{2}}$$