Problem 58 - Calculus for Biology and Medicine

Question

Use either substitution or integration by parts to evaluate each integral. $$ \int \frac{1}{x^{2}+3} d x $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{\sqrt{3}} \tan^{-1}\left( \frac{x}{\sqrt{3}} \right) + C \).

Step 1: Identify the Integral to be Solved

The given integral is \( \int \frac{1}{x^2 + 3} \, dx \). We need to determine the appropriate technique to solve it, whether substitution, integration by parts, or another method.

Step 2: Recognize a Standard Form

The integral \( \int \frac{1}{x^2 + 3} \, dx \) resembles the standard form \( \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C \). For our integral, it fits this form.

Step 3: Identify Parameters for the Formula

In the standard form \( \int \frac{1}{a^2 + x^2} \, dx \), identify \( a = \sqrt{3} \). This means that our integral can be directly computed using this formula.

Step 4: Apply the Arctangent Formula

According to the formula, our integral becomes \( \frac{1}{\sqrt{3}} \tan^{-1}\left( \frac{x}{\sqrt{3}} \right) + C \). This is the evaluation of the integral.

Key Concepts

Substitution Method

Substitution is often used when an integral involves a function and its derivative. The substitution helps simplify the integral into a form that is easier to work with. Imagine you're trying to navigate a complicated road and suddenly you find a shortcut. That's what substitution does in integration. It creates a path to make the task smoother and simpler.Here's a simple way to use the substitution method:

• Identify a part of the integral to substitute with a temporary variable, like \( u \).
• Find the derivative of \( u \) in terms of \( x \), then replace \( dx \) with \( du \).
• Perform the integration with respect to the new variable \( u \).
• Convert back to the original variable by substituting \( u \) back in terms of \( x \).

Even though the provided solution didn't use substitution, understanding it is key for cases where it leads to an easier computation. It opens doors to solving integrals which might otherwise seem complex.

Integration by Parts

Integration by parts is a technique akin to the product rule for differentiation, allowing us to integrate the product of two functions. Remember the joy of dissecting a puzzle into smaller, manageable pieces? That's the beauty of integration by parts - it lets us break down an integral into parts we can handle.The formula used is:\[\int u \, dv = uv - \int v \, du\]To use it, follow these steps:

• Choose \( u \) and \( dv \) from the integral. A common trick is to choose \( u \) from logarithmic, inverse trigonometric, or algebraic functions, and \( dv \) from the rest.
• Differentiate \( u \) to find \( du \) and integrate \( dv \) to find \( v \).
• Apply the integration by parts formula.
• Simplify the result if necessary.

While the exercise didn't require this method, it's indispensable for integrals with products of functions. It's like having a multi-tool in your integration kit.

Arctangent Formula

The arctangent formula is often used when dealing with integrals of the form \( \int \frac{1}{a^2 + x^2} \, dx \). This particular integral looks much like a piece of art fitting so beautifully into its frame.The formula goes as:\[\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C\]Here’s how you can apply this formula:

• Identify the value of \( a \). In this exercise, \( a = \sqrt{3} \).
• Substitute into the formula to solve the integral directly.
• The result becomes: \( \frac{1}{\sqrt{3}} \tan^{-1} \left(\frac{x}{\sqrt{3}} \right) + C \).

Using the arctangent formula offers a direct solution to integrals that seem to defy simplistic substitution or standard algebraic manipulation. It's a powerful tool in handling specific integral forms.