Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{d x}{9 x^{2}+18 x+10}\)

Step-by-Step Solution

Answer

The integral is \(3 \arctan(3(x + 1)) + C\).

Step 1: Identify the Type of Integral

This integral involves a rational function, specifically one with a quadratic denominator. The given integral is \( \int \frac{d x}{9 x^{2}+18 x+10} \). The standard approach to handle this is to complete the square in the quadratic expression.

Step 2: Complete the Square in the Denominator

Rewrite the denominator \(9x^2 + 18x + 10\) to make completing the square easier. First, factor out the 9: \(9(x^2 + 2x) + 10\). Complete the square for \(x^2 + 2x \) by adding and subtracting 1: \(9((x+1)^2 - 1) + 10 = 9(x+1)^2 + 1\). Now the expression becomes \(\int \frac{d x}{9((x+1)^2) + 1}\).

Step 3: Simplify and Substitute

To simplify, consider the substitution \(u = x + 1\), which implies \(du = dx\). The integral becomes \(\int \frac{du}{9u^2 + 1}\). Look for a standard integration formula related to this expression.

Step 4: Apply the Standard Integral Formula

The integration formula for \(\int \frac{du}{a^2 + u^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C \) can be applied here. Compare and recognize that \(a^2 = 1/9\), so \(a = 1/3\). Applying the formula, the integral becomes \(\frac{1}{1/3}\arctan(3u) + C = 3\arctan(3u) + C\).

Step 5: Revert back to Original Variable

Recall that \(u = x + 1\). Substitute back to obtain \(3\arctan(3(x + 1)) + C\) as the solution in terms of the original variable \(x\).

Step 6: Verify the Solution

Verify by differentiating \(3\arctan(3(x + 1)) + C\). Use the chain rule to ensure the derivative provides the original integrand \(\frac{1}{9x^2 + 18x + 10}\). This confirms the integration is correct.

Key Concepts

Rational Functions

Rational functions are expressions that involve the ratio of two polynomials. In calculus, they often appear in integral problems where you need to find the antiderivative or the area under the curve described by such a ratio. A typical rational function takes the form \( \frac{P(x)}{Q(x)} \), where both \(P(x)\) and \(Q(x)\) are polynomials.
A key factor to consider when integrating rational functions is the behavior of the function, especially as the degree of the numerator and the denominator changes. If the denominator is of a higher degree than the numerator, the rational function is proper, and you may use partial fractions or other simplification techniques to find the integral.
If the function is improper, where the numerator’s degree is greater than or equal to that of the denominator, polynomial division might be necessary to simplify the function before integrating. Proper simplification or transformation can reveal more about the function's behavior, which assists in integrating them.

Completing the Square

Completing the square is a technique used in algebra to transform a quadratic expression into a perfect square trinomial plus or minus a constant. It’s notably handy in integration and quadratic equations.
In the context of integration, this method helps when dealing with a quadratic polynomial in the denominator of a rational function. For instance, a quadratic expression like \(9x^2 + 18x + 10\) can be rewritten to an easier-to-integrate form. Here's how it works:

• Factor out the leading coefficient from the terms involving \(x\), in this case, 9.
• Form a perfect square trinomial by adding and subtracting the square of half the coefficient of \(x\), which is \((\frac{18}{18})^2 = 1\).
• Reorganize your expression to reflect these changes, and adjust constants accordingly.

This simplification eases further steps like substitution or applying standard formulas.

Substitution Method

The substitution method in calculus is a powerful tool for transforming a difficult integral into a more manageable one. Often, this involves substituting a part of the function with a new variable to simplify the integrand.
For the integral \( \int \frac{d x}{9x^2 + 18x + 10} \), after completing the square, the substitution \( u = x + 1 \) was made. This choice aligns with the transformed quadratic expression obtained by completing the square.
The substitution technique involves changing both the variable and the differential, leading us to a new integral in terms of \(u\) and \(du\), making it easier to solve. This step is crucial because it transforms the integral into a recognizable form that allows the application of a standard integral formula.
Always remember to convert back to the original variable once the integration is complete to keep the result in the context of the original problem statement.

Integration Formula

Integration formulas help calculate the antiderivatives of functions under various conditions. One frequently used formula is for integrating expressions of the type \( \int \frac{du}{a^2 + u^2} \). Its solution is \( \frac{1}{a} \arctan \left( \frac{u}{a} \right) + C \), where \(a\) is a constant and \(C\) is the integration constant.
In applying this formula to our integral, we identify that the expression fits the pattern with \(a = \frac{1}{3}\) after completing the square and making necessary substitutions. The formula simplifies finding the antiderivative, especially when integrating expressions involving sum or differences of squares.
Using known formulas saves time and provides reliable methods to tackle complex integrals. After applying the formula, substitute back the expressions of the original variable, ensuring all components align with the initial problem setup.