Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x² √(4x - 9))

Verified step by step guidance

1. Identify the integral to solve: \(\int \frac{dx}{x^{2} \sqrt{4x - 9}}\).
2. Look for a suitable substitution to simplify the square root expression. Since the integrand contains \(\sqrt{4x - 9}\), consider the substitution \(t = \sqrt{4x - 9}\) or express \(x\) in terms of \(t\) to simplify the root.
3. Rewrite \(x\) and \(dx\) in terms of \(t\) using the substitution. For example, if \(t = \sqrt{4x - 9}\), then \(t^{2} = 4x - 9\), which implies \(x = \frac{t^{2} + 9}{4}\). Differentiate to find \(dx\) in terms of \(dt\).
4. Substitute \(x\) and \(dx\) back into the integral, and simplify the resulting expression to a form that matches an integral formula from the table of integrals.
5. Use the appropriate integral formula from the table to evaluate the integral in terms of \(t\), then substitute back to express the answer in terms of \(x\).

Key Concepts

Integration Using Standard Integral Forms

Many integrals can be evaluated by recognizing their form and matching them to standard integral formulas found in tables. This approach simplifies complex integrals by avoiding lengthy derivations and directly applying known results.

Substitution Method for Integrals

Substitution involves changing variables to simplify the integral into a standard form. By letting a new variable represent a complicated expression, the integral becomes easier to evaluate using known formulas or techniques.

Handling Integrals Involving Square Roots and Rational Functions

Integrals with expressions like √(ax + b) and rational functions often require algebraic manipulation or substitution to simplify the root and denominator. Recognizing patterns and applying appropriate substitutions is key to solving these integrals.