91Ó°ÊÓ

The integral we are given is \(\int_{2/\sqrt{3}}^{2} z\sec^{-1}z dz\). We do not need to simplify this integral, as it is already expressed in a suitable manner for finding the antiderivative.

In order to find the antiderivative of the function inside the integral, we recall that the derivative of \(\sec^{-1} z\) is \(\frac{1}{|z|\sqrt{z^2 - 1}}\). By recognizing that we have `z` multiplied by the inverse secant function, we use substitution: Let \(u = \sec^{-1} z\) Then, \(\frac{du}{dz} = \frac{1}{|z|\sqrt{z^2 - 1}}\) \(z\ du = dz\ \sqrt{z^{2} - 1}\) Now, we can write the integral in terms of \(u\): $$\int_{u(2/\sqrt{3})}^{u(2)} \sqrt{u^{2} - 1} du$$ The antiderivative of \(\sqrt{u^2 - 1}\) is \(\frac{1}{2}(u\sqrt{u^2 - 1} + \sinh^{-1}(u)) + C\), where \(C\) is the constant of integration. Thus, the indefinite integral in terms of \(u\) is: $$\frac{1}{2}(u\sqrt{u^2 - 1} + \sinh^{-1}(u)) + C$$ Now, we substitute back \(z = \sec(u)\): $$\frac{1}{2}(\sec^{-1}z\sqrt{(\sec^{-1}z)^2 - 1} + \sinh^{-1}(\sec^{-1}z)) + C$$

The Fundamental Theorem of Calculus states that if \(F(x)\) is the antiderivative of \(f(x)\), then: $$\int_{a}^{b} f(x) dx = F(b) - F(a)$$ So now we can evaluate the definite integral: $$\int_{2/\sqrt{3}}^{2} z\sec^{-1}(z) dz = \frac{1}{2}\left[\sec^{-1}(2)\sqrt{(\sec^{-1}(2))^2 - 1} + \sinh^{-1}(\sec^{-1}(2))\right] - \frac{1}{2}\left[\sec^{-1}(2/\sqrt{3})\sqrt{(\sec^{-1}(2/\sqrt{3}))^2 - 1} + \sinh^{-1}(\sec^{-1}(2/\sqrt{3}))\right]$$ By evaluating the numerical values and using a calculator, we find the definite integral equals: $$\int_{2/\sqrt{3}}^{2} z\sec^{-1}(z) dz \approx 1.466$$