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

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

Evaluate

$$\frac{x^{5}-2x^{3}+\pi x^{2}+2x-\pi }{x\left(x^{2}-1\right)}$$

Short Solution Steps

Express x\times \frac{x^{3}}{x^{2}-1} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{x}{x}.

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

Since \frac{\pi }{x} and \frac{2x}{x} have the same denominator, subtract them by subtracting their numerators.

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

Factor x^{2}-1.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x and \left(x-1\right)\left(x+1\right) is x\left(x-1\right)\left(x+1\right). Multiply \frac{\pi -2x}{x} times \frac{\left(x-1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}. Multiply \frac{xx^{3}}{\left(x-1\right)\left(x+1\right)} times \frac{x}{x}.

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

Since \frac{\left(\pi -2x\right)\left(x-1\right)\left(x+1\right)}{x\left(x-1\right)\left(x+1\right)} and \frac{xx^{3}x}{x\left(x-1\right)\left(x+1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(\pi -2x\right)\left(x-1\right)\left(x+1\right)+xx^{3}x.

$$\frac{-\pi +\pi x^{2}-2x^{3}+2x+x^{5}}{x\left(x-1\right)\left(x+1\right)}$$

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

$$\frac{-\pi +\pi x^{2}-2x^{3}+2x+x^{5}}{x^{3}-x}$$

Differentiate w.r.t. x

$$\frac{2x^{7}-4x^{5}-\pi x^{4}+2\pi x^{2}-\pi }{\left(x\left(x^{2}-1\right)\right)^{2}}$$