Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(\int \frac{2 x+1}{x^{2}+2 x+2} d x\)

Step-by-Step Solution

Answer

\( \ln|(x+1)^2+1| - \tan^{-1}(x+1) + C \).

Step 1: Simplify the Denominator

Let's begin by simplifying the denominator of the integrand if possible. The expression in the problem is \(x^2 + 2x + 2\). By completing the square, we can express it as:\[x^2 + 2x + 1 + 1 = (x+1)^2 + 1\]. This allows us to rewrite the integral in a simpler form, using the identity we just found.

Step 2: Express the Numerator Appropriately

Notice that we have \(2x + 1\) in the numerator. It can be split as \((2x + 2) - 1\), allowing us to rewrite the integral as:\[\int \frac{2x+2-1}{(x+1)^2+1} dx = \int \frac{2(x+1)}{(x+1)^2+1} dx - \int \frac{1}{(x+1)^2+1} dx\]. This breaks the integral into two simpler terms.

Step 3: Solve the First Integral Using Substitution

Consider the first integral \(\int \frac{2(x+1)}{(x+1)^2+1} dx\). Let \(u = (x+1)^2+1\), so \(du = 2(x+1) \, dx\). This integral converts to \(\int \frac{1}{u} du\), which is \(\ln|u| + C_1 = \ln|(x+1)^2+1| + C_1\).

Step 4: Solve the Second Integral Using the Arctangent Formula

Now, consider the second integral \(\int \frac{1}{(x+1)^2+1} dx\). This is a standard form related to the arctangent function. It equals to \(\tan^{-1}(x+1) + C_2\).

Step 5: Combine Results and Include Constants

Combine the results of the two integrals obtained: \(\ln|(x+1)^2+1| + C_1 - \tan^{-1}(x+1) + C_2\). Since the constants \(C_1\) and \(C_2\) are arbitrary, we can combine these into a single constant \(C\). Therefore, the final answer is:\[\ln|(x+1)^2+1| - \tan^{-1}(x+1) + C\].

Key Concepts

Completing the Square

Completing the square is a very helpful technique often used in algebra and calculus to simplify quadratic expressions. By transforming the expression into a perfect square trinomial, it can become much easier to work with.

In our original exercise, the denominator of the integrand is given as \(x^2 + 2x + 2\). To complete the square, we begin by focusing on the first two terms \(x^2 + 2x\). The goal is to add and subtract a number in such a way that the expression becomes a perfect square:

• Add and subtract \(1\), so the expression changes to \(x^2 + 2x + 1 - 1 + 2\)
• Now, it becomes \((x + 1)^2 + 1\)

This reformulation allows us to express the quadratic in a simpler form, which is vital for subsequent integration steps.

Substitution Method

The substitution method is a crucial technique in calculus that simplifies integrals by changing the variable of integration. It is akin to changing the perspective on the function to make calculations more manageable.

For the integral \(\int \frac{2(x+1)}{(x+1)^2+1} dx\), a substitution is clearly appropriate. Here, we set:

• \(u = (x+1)^2 + 1\)
• This makes \(du = 2(x+1) \, dx\)

The integral then simplifies to \(\int \frac{1}{u} \, du\), which integrates easily to \(\ln|u| + C_1\). Ultimately, we substitute back \((x+1)^2 + 1\) for \(u\), yielding \(\ln|(x+1)^2 + 1| + C_1\). This method highlights how substitution aligns the structure of the integral with standard forms.

Arctangent Function

The arctangent function, denoted as \(\tan^{-1}(x)\), is a principal inverse of the tangent function. It's highly useful in integration when dealing with expressions of the form \(\frac{1}{x^2 + 1}\).

In our exercise, we face the integral \(\int \frac{1}{(x+1)^2+1} \, dx\). This is a classic setup for the arctangent form:

• The standard integral of \(\int \frac{1}{a^2 + x^2} \, dx\) is \(\frac{1}{a}\tan^{-1}(\frac{x}{a}) + C\)

Here, \(a = 1\) and the form \((x+1)^2\) allows us to directly use this rule, giving us \(\tan^{-1}(x+1) + C_2\). Recognizing these forms speeds up integration considerably.

Integral Simplification

Integral simplification is all about restructuring a complex integral into manageable parts. This involves breaking down the problem into fundamental integrals, each of which is easier to solve individually.

In the given integral, \(\int \frac{2x+1}{x^2+2x+2} \, dx\), after completing the square, we redefine terms to simplify the process:

• Rewrite the numerator as \((2x + 2) - 1\)
• Break down the integral into \(\int \frac{2(x+1)}{(x+1)^2+1} \, dx - \int \frac{1}{(x+1)^2+1} \, dx\)

This allows us to handle each part with appropriate methods, like substitution and recognizing the arctangent form. Thus, solving each part separately simplifies the integral greatly, ultimately leading to the final simplified answer. Remember, integral simplification often requires creative manipulation of terms to align with known integral forms.