Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{1}{x^{2}+2 x+5} d x\)

Step-by-Step Solution

Answer

\( \frac{1}{2} \tan^{-1}\left(\frac{x+1}{2}\right) + C \).

Step 1: Identify the integration form

The integral \( \int \frac{1}{x^{2}+2x+5} \, dx \) resembles the form \( \int \frac{1}{u^{2}+a^{2}} \, du \), which is a standard integral form that results in the inverse tangent function. We aim to transform the integrand into this form by completing the square in the denominator.

Step 2: Complete the square

The expression \( x^2 + 2x + 5 \) can be written by completing the square as \( (x+1)^2 + 4 \). This is achieved by adding and subtracting 1 inside the quadratic term to form a perfect square: \( x^2 + 2x + 1 = (x+1)^2 \), and thus \( (x+1)^2 + 4 \).

Step 3: Substitute to simplify the integral

Let \( u = x + 1 \), so \( du = dx \) and the integral becomes \( \int \frac{1}{u^2 + 4} \, du \).

Step 4: Apply the inverse tangent form

The integral \( \int \frac{1}{u^2 + 4} \, du \) is now of the form \( \int \frac{1}{u^2 + a^2} \, du \) where \( a^2 = 4 \), hence \( a = 2 \). This integral evaluates to \( \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C \). Substituting our values, we get \( \frac{1}{2} \tan^{-1}\left(\frac{u}{2}\right) + C \).

Step 5: Substitute back to original variable

Substitute \( u = x + 1 \) back into the expression to revert to the original variable, giving us the final result: \( \frac{1}{2} \tan^{-1}\left(\frac{x+1}{2}\right) + C \).

Key Concepts

Completing the Square

Completing the square is a useful technique for transforming quadratic expressions into a perfect square trinomial, making them easier to work with. Consider the quadratic expression \(x^2 + 2x + 5\). The goal is to rewrite it as something of the form \((x + h)^2 + k\). Here’s how you do it:

• Identify the coefficient of the linear term (the \(2x\)).
• Take half of the coefficient of \(x\) (which is 2) to get 1, and square it to get 1.
• Add and subtract this value inside the expression: \(x^2 + 2x + 1 - 1 + 5\).
• This simplifies to \((x+1)^2 + 4\).

By completing the square in this way, you can turn complex quadratic expressions into easier forms similar to \(a^2 + b^2\), which are helpful for integration and other calculus problems. This technique also lays the groundwork for using other integration methods effectively.

Inverse Tangent Function

The inverse tangent function, denoted as \(\tan^{-1}(x)\), is a key player in integration techniques, especially when dealing with integrals involving the form \(\frac{1}{u^2 + a^2}\). This arises from the integral \(\int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C\).

• The inverse tangent function is the antiderivative of this specific integral form and maps a range from the real numbers back to angles measured in radians.
• Its derivative is \(\frac{1}{1+x^2}\), making it particularly useful for problems that involve integration of rational functions where the inverse tangent can be applied.

Understanding how and why the inverse tangent function comes into play helps when simplifying integrals and recognizing standard integral forms in calculus problems. It is a very handy function within trigonometric integration scenarios.

Substitution Method

The substitution method is an essential integration technique used to simplify integrals by changing variables. Consider the integral \(\int \frac{1}{x^2 + 2x + 5} \, dx\). After completing the square to get \((x+1)^2 + 4\), substitution steps in:

• Set \(u = x + 1\), making the derivative \(du = dx\).
• Replace \(x+1\) with \(u\) in the integral, transforming it to \(\int \frac{1}{u^2 + 4} \, du\).

This substitution simplifies the integration process by aligning the integral with the form that corresponds to the inverse tangent function. Substitution helps convert complex integrals into forms that are more recognizable and manageable, paving the way for straightforward application of known results or forms.

Standard Integral Forms

Understanding standard integral forms is pivotal for efficiently solving integrals. One common form is \(\int \frac{1}{u^2 + a^2} \, du = \frac{1}{a} \tan^{-1}\left(\frac{u}{a}\right) + C\). Here's why knowing these forms is helpful:

• They allow quick recognition of integral types that should be solved using specific trigonometric or logarithmic functions.
• This reduces the time and effort in deriving results from first principles every time.
• Recognizing a standard integral form can guide the choice of substitution or transformation methods, streamlining the process.

Being familiar with these integral forms can greatly assist in tackling calculus problems efficiently by providing a shortcut to arriving at the solution for integrals that match these forms.