60–69. Completing the square Evaluate the following integrals.
62. ∫ du / (2u² - 12u + 36)

Verified step by step guidance

1. Start by examining the quadratic expression in the denominator: \$2u^{2} - 12u + 36\(. To simplify the integral, factor out the coefficient of \)u^{2}$ from the quadratic terms.
2. Rewrite the denominator as \$2(u^{2} - 6u + 18)$ by factoring out the 2. This will make completing the square inside the parentheses easier.
3. Complete the square for the expression inside the parentheses: \(u^{2} - 6u + 18\). Recall that to complete the square for \(u^{2} - 6u\), add and subtract \((\frac{6}{2})^{2} = 9\) inside the expression.
4. Rewrite \(u^{2} - 6u + 18\) as \((u - 3)^{2} + (18 - 9)\), which simplifies to \((u - 3)^{2} + 9\). Now the denominator is \$2[(u - 3)^{2} + 9]$.
5. Substitute \(w = u - 3\) to simplify the integral to the form \(\int \frac{du}{2(w^{2} + 9)}\). Then, factor out constants and use the standard integral formula for \(\int \frac{dx}{x^{2} + a^{2}}\) to proceed with integration.

Key Concepts

Completing the Square

Completing the square is a technique used to rewrite a quadratic expression in the form ax² + bx + c as a perfect square plus or minus a constant. This simplifies integration by transforming the denominator into a recognizable form, such as (u - h)² + k, which is easier to integrate using standard formulas.

Integration of Rational Functions

Integrating rational functions often involves manipulating the integrand into a simpler form. When the denominator is a quadratic expression, rewriting it by completing the square allows the use of standard integral formulas involving arctangent or logarithmic functions, depending on the form of the quadratic.

Standard Integral Formulas Involving Quadratics

Certain integrals involving quadratics have standard results, such as ∫ dx / (x² + a²) = (1/a) arctan(x/a) + C. Recognizing the denominator as a sum of squares after completing the square enables direct application of these formulas, facilitating the evaluation of the integral.