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

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

Evaluate

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

Short Solution Steps

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}{s^{2}+1} times \frac{s}{s}. Multiply \frac{1}{s} times \frac{s^{2}+1}{s^{2}+1}.

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

Since \frac{ss}{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.

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

Do the multiplications in ss+s^{2}+1.

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

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

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

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

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

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

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

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

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

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

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

Differentiate w.r.t. s

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