1. Evaluate \(\int \frac{\sqrt{4x^2 - 1}}{x^2} \, dx\). Use substitution or integration by parts combined with algebraic manipulation to solve this integral. 2. Consider the parametric equations for the hyperbola given by \(\frac{x^2}{4} - \frac{y^2}{9} = 1\). The correct parametric form uses hyperbolic functions: \(x = 2 \cosh t, y = 3 \sinh t\). 3. Evaluate \(\int e^{-2x} \sinh x \, dx\). Use integration by parts or rewrite \(\sinh x\) in exponential form, then integrate. 4. For the parabola described by \(x^2 - 2x + 8y = 15\), rewrite it in vertex form to find vertex, focus, and directrix. Steps for integral \(\int \frac{\sqrt{4x^2 - 1}}{x^2} \, dx\): 1. Let \(I = \int \frac{\sqrt{4x^2 - 1}}{x^2} \, dx\). 2. Use substitution: let \(u = \sqrt{4x^2 -1}\Rightarrow u^2 = 4x^2 -1\). 3. Differentiate \(u^2\) to find \(du\) in terms of \(dx\). 4. Express everything in terms of \(u\) and simplify. 5. Integrate and back-substitute. Answer: $$ I = 2 \ln|2x + \sqrt{4x^2 -1}| - \frac{\sqrt{4x^2 -1}}{x} + C $$ Steps for parametric equation: 1. Recognize hyperbola standard parametric form uses hyperbolic functions. 2. \(x = 2 \cosh t, y = 3 \sinh t\) satisfy \(\frac{x^2}{4} - \frac{y^2}{9} = 1\). Steps for \(\int e^{-2x} \sinh x \, dx\): 1. Express \(\sinh x = \frac{e^x - e^{-x}}{2}\). 2. Integrate term by term: $$\int e^{-2x} \frac{e^x - e^{-x}}{2} \, dx = \frac{1}{2} \int (e^{-x} - e^{-3x}) \, dx$$ 3. Integrate: $$ \frac{1}{2} \left(-e^{-x} + \frac{e^{-3x}}{3} \right) + C $$ 4. Simplify: $$ -\frac{1}{2} e^{-x} + \frac{1}{6} e^{-3x} + C $$ This matches $$ -\frac{1}{3} [e^{-2x} \cosh x + 2 e^{-2x} \sinh x] + C$$ Steps for parabola \(x^{2} - 2x + 8y = 15\): 1. Complete square for \(x^{2} - 2x\): $$ x^2 - 2x + 1 = (x -1)^2 $$ Rewrite equation: $$ (x -1)^2 + 8y = 16 $$ 2. Rearrange: $$ 8y = 16 - (x -1)^2 \Rightarrow y = 2 - \frac{(x-1)^2}{8} $$ 3. This is a parabola with vertex at \((1, 2)\). 4. The parabola opens downward since coefficient of squared term is negative. 5. Focus and directrix found by \(4p = 8 \Rightarrow p=2\). Focus: \((1, 2 + p) = (1,4)\). Directrix: \(y = 2 - p = 0\). Answer: parabola with vertex (1, 2), focus (1, 4), directrix \(y = 0\). Final answers: Integral: $$2 \ln |2x + \sqrt{4x^2 -1}| - \frac{\sqrt{4x^2 -1}}{x} + C$$ Parametric equations: \(x=2 \cosh t, y=3 \sinh t\). Exponential integral: $$-\frac{1}{3} [e^{-2x} \cosh x + 2 e^{-2x} \sinh x] + C$$ Parabola with vertex at (1, 2), focus (1, 4), directrix \(y=0\).