79–82. {Use of Tech} Double table look-up The following integrals may require more than one table look-up. Evaluate the integrals using a table of integrals, and then check your answer with a computer algebra system.
82. ∫ (sin⁻¹(ax)) / x² dx, a > 0

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{\sin^{-1}(ax)}{x^{2}} \, dx\), where \(a > 0\).
2. Consider using integration by parts, since the integrand is a product of functions involving an inverse trigonometric function and a power of \(x\). Let \(u = \sin^{-1}(ax)\) and \(dv = \frac{1}{x^{2}} dx\).
3. Compute the derivatives and integrals needed for integration by parts: find \(du = \frac{a}{\sqrt{1 - a^{2}x^{2}}} dx\) and \(v = -\frac{1}{x}\).
4. Apply the integration by parts formula: \(\int u \, dv = uv - \int v \, du\), substituting the expressions for \(u\), \(v\), and \(du\).
5. Simplify the resulting integral, which will likely require a second table look-up or substitution to evaluate \(\int \frac{1}{x \sqrt{1 - a^{2}x^{2}}} dx\). Use a table of integrals or a computer algebra system to find this integral.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, like sin⁻¹(x), are the inverses of the standard trigonometric functions and return an angle whose sine is x. Understanding their properties and derivatives is essential for integrating expressions involving these functions.

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation. It is useful when integrating products of functions, such as an inverse trigonometric function multiplied by an algebraic expression, by reducing the integral into simpler parts.

Use of Integral Tables and Computer Algebra Systems

Integral tables provide standard forms of integrals that can simplify complex integration problems. Computer algebra systems (CAS) help verify solutions and handle complicated integrals, ensuring accuracy and saving time.