Problem 33 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int \frac{y}{y^{2}+4} d y $$

Step-by-Step Solution

Answer

The integral is \( \frac{1}{2} \ln |y^2 + 4| + C \).

Step 1: Identify Integration by Substitution

The integral \( \int \frac{y}{y^{2}+4} \, d y \) can be solved using substitution. Here, you notice that the derivative of the denominator \( y^2 + 4 \) is similar to the numerator. This suggests using substitution.

Step 2: Perform Substitution

Let \( u = y^2 + 4 \). Then, take the derivative of \( u \) with respect to \( y \), giving us \( \frac{du}{dy} = 2y \). Therefore, \( du = 2y \, dy \). Our integral becomes \( \int \frac{1}{2} \frac{1}{u} \, du \).

Step 3: Integrate Using Basic Logarithmic Rule

The integral \( \int \frac{1}{u} \, du \) is known to be \( \ln |u| + C \), where \( C \) is the constant of integration. So, \( \int \frac{1}{2} \frac{1}{u} \, du = \frac{1}{2} \ln |u| + C \).

Step 4: Substitute Back the Original Variable

Replace \( u \) with the expression in terms of \( y \). Thus, we have \( \frac{1}{2} \ln |y^2 + 4| + C \).

Step 5: Confirm by Differentiation

Differentiate the result \( \frac{1}{2} \ln |y^2 + 4| + C \) to check for the original integrand, \( \frac{y}{y^2+4} \). \[ \frac{d}{dy} \left( \frac{1}{2} \ln |y^2 + 4| \right) = \frac{1}{2} \cdot \frac{2y}{y^2 + 4} = \frac{y}{y^2 + 4} \]. This verifies the correctness of the integration.

Key Concepts

Integration by Substitution

Integration by substitution is a powerful method used to evaluate integrals. It essentially involves changing variables to simplify the integration process. When you look at an integral like \( \int \frac{y}{y^{2}+4} \, d y \), it's crucial to recognize patterns. If the derivative of one part of the integrand matches another part, substitution might be useful.
\( \textbf{Steps for Substitution:} \)

• Identify the substitution: For our example, choose \( u = y^2 + 4 \).
• Find \( du \): The differentiation with respect to \( y \) gives \( \frac{du}{dy} = 2y \).
• Express \( dy \) in terms of \( du \): We get \( du = 2y \, dy \) or \( dy = \frac{1}{2y} \, du \).
• Convert the integral: Replace parts of the integral with \( u \) and \( du \), simplifying to \( \int \frac{1}{2} \frac{1}{u} \, du \).

Substitution transforms a complex integral into an easier one that can often be solved using basic rules.

Logarithmic Integration

Logarithmic integration occurs when dealing with integrals involving the natural log function. The integral \( \int \frac{1}{u} \, du \) is the prime example.
When performing substitution, you might end up with an integral in the form of \( \int \frac{1}{u} \, du \), which integrates to \( \ln|u| + C \). Substitution made our integral \( \int \frac{1}{2} \frac{1}{u} \, du \), resulting in \( \frac{1}{2} \ln|u| + C \).
This manipulation turns the integration into a simple logarithmic expression. Hence, identifying this structure and using logarithmic rules simplifies the process, allowing us to find the solution efficiently.

Differentiation Verification

Differentiation verification is a crucial step in confirming the correctness of an integration process. Once you've derived the antiderivative, you should differentiate it to see if it yields the original integrand.
Let's consider the integral result \( \frac{1}{2} \ln|y^2 + 4| + C \). By differentiating this expression with respect to \( y \), you confirm its correctness. Using the chain rule:

• Differentiate the logarithmic part: \( \frac{d}{dy} \left( \frac{1}{2} \ln |y^2 + 4| \right) \) results in \( \frac{1}{2} \cdot \frac{2y}{y^2 + 4} \).
• Simplification: This expression simplifies back to \( \frac{y}{y^2 + 4} \).

Thus, differentiation shows you've reverted to the original integrand, proving the solution's validity. It's like a mathematical checkpoint ensuring that your integration was executed correctly.