Exercise 12

Consider the indefinite integral $\int x \sqrt{x-1} \, dx$.

1. At first glance, this integrand may not seem suited to substitution due to the presence of $x$ in separate locations in the integrand. Nonetheless, using the composite function $\sqrt{x-1}$ as a guide, let $u = x-1$. Determine expressions for both $x$ and $dx$ in terms of $u$.

2. Convert the given integral in $x$ to a new integral in $u$.

3. Evaluate the integral in (b) by noting that $\sqrt{u} = u^{1/2}$ and observing that it is now possible to rewrite the integrand in $u$ by expanding through multiplication.

4. Evaluate each of the integrals $\int x^2 \sqrt{x-1} \, dx$ and $\int x \sqrt{x^2 - 1} \, dx$. Write a paragraph to discuss the similarities among the three indefinite integrals in this problem and the role of substitution and algebraic rearrangement in each.