Problem 62 - Essential Calculus

Question

Verify each rule by differentiating. Let \(a>0\). $$ \int \frac{d u}{a^{2}+u^{2}}=\frac{1}{a} \arctan \frac{u}{a}+C $$

Step-by-Step Solution

Answer

The derivative of \(\frac{1}{a} \arctan(\frac{u}{a}) + C\) is indeed \(\frac{1}{a^{2}+u^{2}}\), which verifies that the integral formula is correct.

Step 1: Prepare the right side of the equation for differentiation

The right side of the equation is \( \frac{1}{a} \arctan \frac{u}{a}+C \). We'll differentiate this with respect to \(u\).

Step 2: Apply the Constant Multiple Rule

Differentiation of the first part, that is, \( \frac{1}{a} \arctan \frac{u}{a} \). We take out \(\frac{1}{a}\) from the derivate because of the constant multiplication rule in differentiation, which means the derivative of a constant times a function is the constant times the derivative of the function. So, it becomes \(\frac{1}{a} (\frac{d}{du} \arctan \frac{u}{a})\). The derivative of the constant \(C\) which is the second part, is 0, because the derivative of a constant is always zero.

Step 3: Use the Derivative Rule for Inverse Tangent

Now, we have to differentiate the inverse tangent function. The derivative of inverse tangent of \(x\) is \(\frac{1}{1+x^{2}}\), so the derivative of \(\arctan \frac{u}{a}\) will be \(\frac{1}{1+(\frac{u}{a})^{2}}\). Hence, the derivative becomes \( \frac{1}{a} \cdot \frac{1}{1+(\frac{u}{a})^{2}}\).

Step 4: Simplify the expression

We can simplify this expression by multiplying both numerator and denominator by \(a^{2}\), resulting in \(\frac{1}{a^{2}+u^{2}}\), which is the integrand on the left. Thus, the derivative of the given expression equals the integrand on the left, demonstrating that the differentiation was performed successfully

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function. In simple terms, it tells us how a function changes as its input changes. Let's say you have a function, and you need to find its rate of change at any given point. Differentiation will help us achieve that by providing a derivative, which is essentially a new function that tells us the slope of the original function at any point.
Here, we applied differentiation to verify the integration formula given in the exercise. By differentiating the right side of the equation, we can check if it matches the integrand on the left side. If both are equal, it confirms that our integration was performed correctly, which acts as a form of verification for the solution.

Constant Multiple Rule

The Constant Multiple Rule in differentiation states that the derivative of a constant times a function is simply the constant times the derivative of the function itself. This is a powerful property that simplifies the differentiation process significantly.
Imagine you have a function like \( c \cdot f(x) \), where \( c \) is a constant. Instead of complicating your process by involving \( c \) heavily in your derivative calculations, you can easily pull out \( c \) and focus on differentiating just \( f(x) \). This makes our job easier and cleaner.

• Example: if \( f(x) = 3x^2 \), applying the Constant Multiple Rule gives us \( 3 \cdot (2x) = 6x \).

In the exercise, we applied this rule to the expression \( \frac{1}{a} \arctan \frac{u}{a} \). By treating \( \frac{1}{a} \) as a constant, we could simplify the differentiation process of the inverse tangent function.

Derivative of Inverse Tangent

To find the derivative of the inverse tangent function, also known as \( \arctan(x) \), we use a specific derivative formula. This formula states that the derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \).
When the argument of the inverse tangent is not just \( x \), like in \( \arctan \frac{u}{a} \), we apply the chain rule alongside the specific formula. The chain rule allows us to find the derivative of composite functions efficiently.

• In our exercise, the chain rule is used following the Constant Multiple Rule, resulting in \( \frac{1}{1+\left(\frac{u}{a}\right)^2} \).

Understanding these rules allows us to solve derivatives involving inverse tangent easily, which forms a major part of verifying integration in our problem.

Integral Verification

Integral verification is the process of confirming that the integration was done correctly. One way to do this is by differentiating the result of the integral and checking if it matches the original integrand.
In our exercise, after performing the integration, we verified it by differentiating the result – \( \frac{1}{a} \arctan \frac{u}{a} + C \). By showing that the derivative of this entire expression gives back the original \( \frac{1}{a^2 + u^2} \), we confirm the integrity of our integration process.

• This method shows the symmetrical relationship between integration and differentiation – they are essentially inverse operations.
• Verification is important to ensure accuracy in solving calculus problems, making it an essential skill for students.

Through integral verification, we gain both confidence and understanding of our mathematical solutions.