Problem 85 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \frac{1}{x^{2}+4} d x $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Integration Technique

The integral \( \int \frac{1}{x^2 + 4} \, dx \) can be solved using trigonometric substitution or recognizing it as a form that involves an arctangent function. This is because the denominator is a sum of squares.

Step 2: Use the Standard Form for Arctangent

The integral \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \) is useful here. In our case, we have \( a = 2 \).

Step 3: Apply the Formula

Substitute \( a = 2 \) into the formula: \[ \int \frac{1}{x^2 + 4} \, dx = \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + C \].

Step 4: Write the Final Solution

Thus, the indefinite integral is \( \int \frac{1}{x^2 + 4} \, dx = \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + C \).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a strategic technique to simplify the integration of functions involving quadratic expressions like \( x^2 + a^2 \). It uses trigonometric identities to transform the integral into a more manageable form. You often encounter this when dealing with integrals having sums or differences of squares.
To apply trigonometric substitution, you select a trigonometric identity that fits well with the expression inside the integral. Typically, you would use:

• \( x = a \tan(\theta) \) if you have \( x^2 + a^2 \)
  
• \( x = a \sin(\theta) \) if you have \( x^2 - a^2 \)
  
• \( x = a \sec(\theta) \) if you have \( a^2 - x^2 \)
  

These substitutions transform the integrand into a function of the trigonometric angle \( \theta \), which is generally easier to integrate. After integration, you use inverse trigonometric functions to switch back to the original variable.

Integration Technique

Integration techniques are fundamental tools in calculus used to evaluate integrals, either definite or indefinite. Different integrals require different methods based on their structure. Some commonly employed techniques include substitution, integration by parts, and partial fraction decomposition.
In this particular exercise, the integral \( \int \frac{1}{x^2 + 4} \, dx \) is effectively evaluated using the arctangent function, which is a special case of integrals involving trigonometric substitution. Recognizing that the expression \( x^2 + 4 \) is of the form \( x^2 + a^2 \), you can use the arctangent formula to find the indefinite integral easily. Understanding when to use specific techniques reduces complex problems into more straightforward computations, making the calculus journey smoother.
Mastering these techniques broadens your ability to tackle a wide array of integrals in your mathematical and scientific endeavors.

Arctangent Function

The arctangent function, often represented as \( \tan^{-1}(x) \), is the inverse of the tangent function and plays a crucial role in calculative methods like integration. In integrals, particularly those involving sums of squares like \( x^2 + a^2 \), recognizing its presence can greatly simplify the problem.
For the integral \( \int \frac{1}{x^2 + a^2} \, dx \), it matches neatly with the formula: \[\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C\]This formula is powerful because it allows for a straightforward evaluation of the integral in just a single step, avoiding more cumbersome calculations. In the original step-by-step solution, substituting \( a = 2 \) leads directly to the result \( \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right) + C \), providing a constant reminder of how versatile the arctangent function is in calculus. Understanding its application helps in recognizing patterns in problem-solving and enhances integration skills.