Problem 24 - Calculus for Biology and Medicine

Question

In Problems , complete the square in the denominator and evaluate the integral. $$ \int \frac{1}{x^{2}+4 x+5} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(\arctan(x+2) + C\).

Step 1: Identify the Quadratic Expression

In the denominator of the integral, we have the quadratic expression \(x^2 + 4x + 5\). To complete the square, we first need to express this quadratic in a perfect square form plus a constant.

Step 2: Complete the Square

Rewrite the quadratic expression \(x^2 + 4x + 5\) by completing the square:1. Take the coefficient of \(x\), which is \(4\), halve it to get \(2\), and then square it to get \(4\).2. Add and subtract this square inside the expression: \[x^2 + 4x + 5 = (x^2 + 4x + 4) + 1 = (x + 2)^2 + 1\].Thus, the expression can be rewritten using the completed square form \((x + 2)^2 + 1\).

Step 3: Rewrite the Integral

Now that we have completed the square, rewrite the integral as:\[\int \frac{1}{(x+2)^2 + 1} \, dx\].This expression fits the standard integral form that resembles \(\arctan(x)\).

Step 4: Apply the Arctangent Formula

The integral \(\int \frac{1}{x^2 + a^2} \, dx\) results in \(\frac{1}{a}\arctan\left(\frac{x}{a}\right) + C\), where \(a = 1\) in our case.Rewriting the integral, we have:\[\int \frac{1}{(x+2)^2 + 1} \, dx = \arctan(x+2) + C\].

Step 5: Evaluate the Integral

Thus, the integral evaluates to \(\arctan(x+2) + C\), where \(C\) is the constant of integration.

Key Concepts

Completing the Square

Completing the square is a fundamental technique in algebra that allows us to rewrite a quadratic expression in the form of \(a(x+b)^2 + c\). This transformation is particularly beneficial when simplifying expressions and solving quadratic equations. In the context of integration, completing the square is invaluable for evaluating integrals involving quadratic expressions in the denominator.

Here's how to do it:

• Identify the quadratic expression, such as \(x^2 + 4x + 5\).
• Take the coefficient of the linear term, halve it, and then square it. For \(4x\), halve \(4\) to get \(2\) and square it to obtain \(4\).
• Add and subtract this square in the expression: \(x^2 + 4x + 5 = (x+2)^2 + 1\).

This step-by-step transformation results in a simplified form that is much easier to handle in integration, especially when dealing with integrals resembling the form of known trigonometric derivative formulas.

Arctangent Integration

Arctangent integration is a common technique for solving integrals that involve a sum of squares in the denominator. Specifically, the integral \( \int \frac{1}{x^2 + a^2} \, dx\) can be evaluated using the arctangent function. Understanding this integration involves recognizing the integral's similarity to the derivative of the arctangent function.

To apply this:

• Ensure the denominator is in the form of \(x^2 + a^2\), where \(a\) is a constant. This often involves completing the square first.
• Recognize that the formula for integration is \frac{1}{a}\arctan(\frac{x}{a}) + C\.

In our specific example, the integral \(\int \frac{1}{(x+2)^2 + 1} \, dx\) simplifies perfectly to \arctan(x+2) + C\. Understanding how to apply this formula is essential for efficiently solving integrals of this complex form.

Definite and Indefinite Integrals

Integrals can be categorized into two types: definite and indefinite. Both play a crucial role in calculus, but they serve different purposes.

An indefinite integral, like the one in our example, represents a family of functions and includes a constant of integration, \(C\). It symbolizes the antiderivative of a function, which means finding a function whose derivative is the integrand. For example, \( \int \frac{1}{(x+2)^2 + 1} \, dx = \arctan(x+2) + C\) represents an indefinite integral.

In contrast, a definite integral calculates the area under a curve from one point to another, resulting in a numerical value. It has specific upper and lower limits. While our current exercise involves indefinite integration, understanding both types helps in recognizing when to apply each process.

• Indefinite integrals have no limits and include \(C\).
• Definite integrals are evaluated over an interval, providing a specific area value.

Mastering both definite and indefinite integrals is essential for solving a wide array of problems in calculus, ranging from determining area under a curve to finding antiderivatives in equations.