Problem 18 - Calculus: One and Several Variables

Question

Calculate. $$\int \frac{d x}{x^{2}+2 x+2}$$.

Step-by-Step Solution

Answer

\(\int \frac{dx}{x^2 + 2x + 2} = \arctan(x + 1) + C\)

Step 1: Complete the square

We rewrite the denominator in the form of a complete square. We have: \(x^2 + 2x + 2 = (x + 1)^2 + 1\) Now the integral becomes: \(\int \frac{dx}{(x + 1)^2 + 1}\)

Step 2: Use substitution

Let's use substitution: let \(u = x + 1\), then \(du = dx\), and the integral becomes: \(\int \frac{du}{u^2 + 1}\)

Step 3: Evaluate the integral

To evaluate this integral, we can recognize it as a standard integral for the arctangent function. More specifically, we have: \(\int \frac{du}{u^2 + 1} = \arctan(u) + C\)

Step 4: Replace the substitution variable

Finally, we replace the substitution variable \(u\) with the original variable \(x\): \(\arctan(u) + C = \arctan(x + 1) + C\) So, the final answer is: \(\int \frac{dx}{x^2 + 2x + 2} = \arctan(x + 1) + C\)

Key Concepts

Substitution Method

The substitution method is a powerful tool in integral calculus that transforms an integral into a simpler form. This method involves changing variables to make integration more manageable. In the given exercise, the substitution used is straightforward: letting \( u = x + 1 \). By doing this, the integral

• \( \int \frac{dx}{(x + 1)^2 + 1} \)

becomes

• \( \int \frac{du}{u^2 + 1} \).

Notice how we replaced 'dx' with 'du' based on the derivative of our substitution rule \( du = dx \). This change allows us to focus on an easier integral form that is often recognizable in standard integral tables.
The ultimate goal of substitution is to simplify the expression to an integrable form, often by reducing polynomial degrees or exposing a trigonometric identity. Once the integral is solved, it's key to substitute back to the original variable (from \( u \) back to \( x \) in this case). This step ensures that the solution matches the original problem context.

Completing the Square

Completing the square is a technique commonly used to simplify quadratic expressions. It helps transform a quadratic term into a perfect square trinomial, making it easier to integrate, especially when combined with substitution. In this exercise, we start with the expression

• \( x^2 + 2x + 2 \).

By rearranging it, we complete the square as follows:

• \( (x + 1)^2 + 1 \).

This format is advantageous because it often matches up with the form equations take in standard integral solutions. Completing the square changes the integral into something we are more familiar with, such as a form based on the Pythagorean identity.
This transformation is crucial when the denominator of the integrand is a quadratic expression, because it can lead to valuable insights and make the problem easier to solve using standard integration techniques and results like the arctangent integral.

Arctangent Function

The arctangent function, denoted \( \arctan(x) \), is the inverse of the tangent function. It is important in calculus due to its relationship with various integral solutions. In this exercise, after applying substitution and completing the square, the final integral happens to be of the form:

• \( \int \frac{du}{u^2 + 1} \).

This is a classic integral form whose solution is \( \arctan(u) + C \). The presence of \( u^2 + 1 \) under the integral sign is a cue to think of the arctangent due to its connection to the derivative \( \frac{d}{du} \arctan(u) = \frac{1}{1+u^2} \).
The arctangent function results from integrating functions that resemble trigonometric identities. In practice, recognizing this link helps solve integrals quickly using standard results.
Ultimately, once we solve using the arctangent, it is essential to convert back to the original variable (e.g., substituting back from \( u \) to \( x+1 \)), completing the integration process.