Integral of 1 over y cubed times square root of y squared minus 1

Find the integral $\displaystyle \int \frac{1}{y^3 \sqrt{y^2 - 1}} \, dy$.

This integral challenges us to manipulate expressions thoughtfully, calling upon our understanding of algebraic manipulation and strategic substitution. A primary strategy in this problem is to simplify the integration process by rewriting or substituting parts of the function. Often, integrals in this form suggest the method of trigonometric substitution, as the presence of a square root involving a difference of squares aligns with trigonometric identities. Trigonometric substitution can simplify expressions involving square roots, particularly when dealing with quadratic expressions. In this integral, consider y = sec(θ), utilizing the relationship $\sec^2(\theta) - 1 = \tan^2(\theta)$. This substitution transforms the integral into a more workable trigonometric form, making it easier to handle and integrate using standard techniques.

Understanding such techniques extends beyond this single problem, forming a component of a larger strategy for handling integrals that appear complex due to their algebraic structure. Mastering this approach involves recognizing which substitutions will simplify the expressions you are working with and knowing the standard integral forms that can result from these transformations. Additionally, these problems skillfully show how trigonometric and inverse trigonometric functions intertwine with integration, deepening your understanding of the connections between different areas of mathematics.

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