What is the integral of (1-z^{-2})sqrt(z+z^{-1)} ?

The integral of (1-z^{-2})sqrt(z+z^{-1)} is 2/3 ((z^2+1)/z)^{3/2}+C

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integral of (1-z^{-2})sqrt(z+z^{-1)}

\int (1 - z^{- 2}) z + z^{- 1} d z

\frac{2}{3} (\frac{z ^{2} + 1}{z})^{\frac{3}{2}} + C

Solution steps

\int (1 - z^{- 2}) z + z^{- 1} d z

Apply u-substitution

= \int u d u

Apply the Power Rule

= \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = z + z^{- 1}

= \frac{2}{3} (z + z^{- 1})^{\frac{3}{2}}

(z + z^{- 1})^{\frac{3}{2}} = (\frac{z ^{2} + 1}{z})^{\frac{3}{2}}

= \frac{2}{3} (\frac{z ^{2} + 1}{z})^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{3} (\frac{z ^{2} + 1}{z})^{\frac{3}{2}} + C

What is the integral of (1-z^{-2})sqrt(z+z^{-1)} ?
The integral of (1-z^{-2})sqrt(z+z^{-1)} is 2/3 ((z^2+1)/z)^{3/2}+C