1. Let's start by stating the problem: We want to rewrite and simplify the integral expression involving \( \frac{\sqrt{t^2-1}}{t} - t^2 \). 2. The expression is \( \frac{\sqrt{t^2-1}}{t} - t^2 \). We want to see if any terms can be simplified or canceled. 3. Notice that the first term is \( \frac{\sqrt{t^2-1}}{t} \). Since \( t \) is in the denominator, we cannot cancel \( t \) inside the square root with the denominator directly. 4. The second term is \( t^2 \), which is separate and cannot be combined with the first term unless we find a common denominator. 5. To combine the terms, write \( t^2 \) as \( \frac{t^3}{t} \) to have a common denominator \( t \): $$ \frac{\sqrt{t^2-1}}{t} - t^2 = \frac{\sqrt{t^2-1}}{t} - \frac{t^3}{t} = \frac{\sqrt{t^2-1} - t^3}{t} $$ 6. This is the simplified form of the expression inside the integral. 7. Important note: You cannot cancel \( t \) inside the square root with the denominator because \( \sqrt{t^2-1} \) is not equal to \( t \sqrt{1 - \frac{1}{t^2}} \) in a way that allows cancellation without changing the expression. 8. Therefore, the integral should be rewritten as: $$ \int \frac{\sqrt{t^2-1} - t^3}{t} \, dt $$ This respects the algebraic rules and avoids incorrect cancellation. Final answer: The integral expression rewritten is \( \int \frac{\sqrt{t^2-1} - t^3}{t} \, dt \).