Problem 25 - Calculus Early Transcendentals

Question

Use a table of integrals to determine the following indefinite integrals. These integrals require preliminary work, such as completing the square or changing variables, before they can be found in a table. $$\int \frac{d x}{x^{2}+2 x+10}$$

Step-by-Step Solution

Answer

Answer: The indefinite integral of the given function is $$\int \frac{d x}{x^{2}+2 x+10} = \frac{1}{3} \arctan\left(\frac{x + 1}{3}\right) + C$$, where \(C\) is the constant of integration.

Step 1: Rewrite the given integral

Start with the given integral: $$\int \frac{d x}{x^{2}+2 x+10}$$

Step 2: Complete the square

Now, let's complete the square for the denominator. The objective is to represent the denominator as \((x+a)^2 + b^2\), where \(a\) and \(b\) are constants. To achieve this, we need to find the correct values for \(a\) and \(b\): Observe that \((x+1)^2 = x^2 + 2x + 1\). To obtain the constant term \(10\) in our denominator, we need to add \(9\) to \((x+1)^2\). That means, \(a = 1\) and \(b = 3\). So, the denominator can be rewritten as: $$x^2 + 2x + 10 = (x+1)^2 + 3^2$$ The integral becomes: $$\int \frac{d x}{(x+1)^2 + 3^2}$$

Step 3: Perform a variable substitution

Choose the substitution \(u = x + 1\). The differential \(d u\) is equal to \(d x\). The integral is now: $$\int \frac{d u}{u^2 + 3^2}$$

Step 4: Use a table of integrals

Now, we can look for this integral in a table of integrals. The table of integrals should have a formula for the following integral: $$\int \frac{d u}{u^2 + a^2}$$ This formula is: $$\int \frac{d u}{u^2 + a^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C$$ Where \(C\) is the constant of integration. Applying this formula to our integral, with \(a = 3\), we get: $$\int \frac{d u}{u^2 + 3^2} = \frac{1}{3} \arctan\left(\frac{u}{3}\right) + C$$ Now replace the substitution, \(u = x + 1\): $$\int \frac{d x}{x^2 + 2x + 10} = \frac{1}{3} \arctan\left(\frac{x + 1}{3}\right) + C$$ Thus, the indefinite integral is: $$\int \frac{d x}{x^2 + 2x + 10} = \frac{1}{3} \arctan\left(\frac{x + 1}{3}\right) + C$$

Key Concepts

Indefinite Integral

The indefinite integral, often denoted as \( \int f(x) \, dx \), represents a family of functions that differ by a constant. It is the reverse process of differentiation. In simpler terms, if you differentiate an indefinite integral, you'll get the original function back. This type of integral does not have limits and represents an antiderivative of a function.
An intuitive way to understand indefinite integrals is to think of them as the cumulative sum of area under the curve of a function, without specific starting or ending points. As each antiderivative can differ by a constant, the result of an indefinite integral always includes '+ C', which stands for the constant of integration.

Completing the Square

Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This is particularly useful when dealing with integrals, as it simplifies the expression and makes it easier to match with standard forms found in a table of integrals.
In our exercise, we started with the expression \( x^2 + 2x + 10 \). To complete the square, we focus on the \( x^2 + 2x \) part. By observing that \((x+1)^2 = x^2 + 2x + 1\), we see we need to add 9 to achieve a perfect square in the original expression. Thus, the expression becomes \((x+1)^2 + 3^2\). This completed square form is simpler to integrate using known formulas.

Variable Substitution

Variable substitution, or change of variables, is a method used to simplify an integral by introducing a new variable. This technique involves substituting a part of the integral with a new variable, allowing us to work with a simpler expression.
In the given problem, after completing the square, we substituted \( u = x + 1 \). The differential \( du \) matches \( dx \), which simplifies the integration process. Now, the integral \( \int \frac{dx}{(x+1)^2 + 3^2} \) becomes \( \int \frac{du}{u^2 + 3^2} \). This substitution step is crucial for fitting the integral into a standard form from a table of integrals.

Table of Integrals

A table of integrals is a collection of standard integral formulas that allow for quick calculation of integrals without going through the integration process each time. These tables include common forms and their solutions.
In our problem, once we've rewritten the integral using substitution, it matches the form \( \int \frac{du}{u^2 + a^2} \). The table of integrals provides the formula \( \int \frac{du}{u^2 + a^2} = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C \). Using this formula, with \( a = 3 \), we can directly find the integrated function, which simplifies and accelerates the solution process.