Solve s/s^2+1+1+1/s+1

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

Evaluate

$$\frac{2s^{3}+2s^{2}+2s+1}{s\left(s^{2}+1\right)}$$

Short Solution Steps

Add 1 and 1 to get 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{s^{2}+1}{s^{2}+1}.

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

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

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

Do the multiplications in s+2\left(s^{2}+1\right).

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

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

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

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

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

Do the multiplications in \left(s+2s^{2}+2\right)s+s^{2}+1.

$$\frac{s^{2}+2s^{3}+2s+s^{2}+1}{s\left(s^{2}+1\right)}$$

Combine like terms in s^{2}+2s^{3}+2s+s^{2}+1.

$$\frac{2s^{2}+2s^{3}+2s+1}{s\left(s^{2}+1\right)}$$

Expand s\left(s^{2}+1\right).

$$\frac{2s^{2}+2s^{3}+2s+1}{s^{3}+s}$$

Differentiate w.r.t. s

$$-\frac{2s^{4}+s^{2}+1}{\left(s\left(s^{2}+1\right)\right)^{2}}$$