Question

Use tables to evaluate the integrals. You may need to complete the square or change variables to put the integral into a form given in the table. $$ \int \frac{d x}{x^{2}+2 x+10} $$

Short answer

\( \int \frac{d x}{x^{2}+2 x+10} = \frac{1}{3} \tan^{-1}\left(\frac{x+1}{3}\right) + C \).

Step-by-step solution

Recognize the Form

The integral \( \int \frac{d x}{x^{2}+2x+10} \) can potentially be evaluated using a table, but it needs to be put into a recognizable form, such as those of standard integrals. This involves completing the square for the quadratic expression \( x^2 + 2x + 10 \).

Complete the Square

Complete the square for the expression \( x^2 + 2x + 10 \). We do this by rewriting it as \( (x+1)^2 + 9 \). Here's the breakdown:- Take half of the coefficient of \( x \) (which is 2) and square it: \( (\frac{2}{2})^2 = 1 \).- Add and subtract this value inside the expression: \( x^2 + 2x + 1 - 1 + 10 = (x+1)^2 + 9 \).

Rewrite the Integral

Substitute the completed square into the integral: \( \int \frac{d x}{(x+1)^2 + 9} \).

Use Standard Integral Form

Recognize that this integral fits the form \( \int \frac{d x}{(x-a)^2 + b^2} = \frac{1}{b} \cdot \tan^{-1}\left(\frac{x-a}{b}\right) + C \). Here, \( a = -1 \) and \( b = 3 \).

Evaluate the Integral

Apply the formula: \( \int \frac{d x}{(x+1)^2 + 9} = \frac{1}{3} \tan^{-1}\left(\frac{x+1}{3}\right) + C \). This is the evaluated integral.

Key concepts

Completing the Square

Completing the square is a helpful algebraic technique that transforms quadratic expressions into perfect squares. This makes it easier to work with integrals involving quadratics. For instance, consider the expression \( x^2 + 2x + 10 \). We need to rewrite this in the form of \( (x+h)^2 + k \).
Here's how you can do it:

• Identify the coefficient of the linear term (here it's 2 in \(2x\)).
• Take half of this coefficient, which is \( \frac{2}{2} = 1 \), and square it to get 1.
• Add and subtract this squared term within the expression: \( x^2 + 2x + 1 - 1 + 10 \).
• You end up with \( (x+1)^2 + 9 \) after simplifying.

By completing the square, the expression becomes cleaner, which is crucial for moving on to standard integral forms. It turns the integral problem into something recognizable and easier to solve.

Standard Integral Forms

Standard integral forms are pre-identified patterns that provide straightforward solutions to common integral types. When integrating functions, recognizing these forms can significantly reduce complexity.
For the integral \( \int \frac{d x}{x^{2}+2x+10} \), completing the square simplified the expression into \( \int \frac{d x}{(x+1)^2 + 9} \). Here, the standard form we use is:

• \( \int \frac{d x}{(x-a)^2 + b^2} = \frac{1}{b} \cdot \tan^{-1}\left(\frac{x-a}{b}\right) + C \)

Recognizing this form is vital for directly applying the formula without having to compute the integral from scratch. It saves both time and effort, especially in more advanced calculus problem-solving scenarios.

Calculus Problem-Solving

Calculus problem-solving often involves strategies for simplifying and transforming expressions. It requires recognizing when and how to use algebraic techniques, like completing the square, and applying formulas from a reference table of integrals.
In solving integrals like \( \int \frac{d x}{x^2 + 2x + 10} \), the problem-solving process includes:

• Transforming the quadratic expression into a complete square.
• Recognizing patterns or forms that match with standard integral formulas.
• Substituting and rearranging expressions into solvable formats.

This systematic approach helps to tackle integrals that at first glance, may look challenging but become manageable with the right tools and techniques.

Definite and Indefinite Integrals

Understanding the difference between definite and indefinite integrals is crucial in calculus. Both have their unique purposes and applications.
- **Definite Integrals** are used to calculate the accumulated quantity, like area under a curve, and have limits of integration (from \(a\) to \(b\)). They result in a numerical value.- **Indefinite Integrals**, on the other hand, do not have limits of integration and thus represent a family of functions, plus a constant \(C\).In the exercise we worked through, the integral \( \int \frac{d x}{x^2 + 2x + 10} \) was indefinite. The goal was to simplify and find its antiderivative, resulting in an expression \( \frac{1}{3} \tan^{-1}\left(\frac{x+1}{3}\right) + C \). This represents all possible antiderivatives of the function.