Problem 72 - University Calculus: Early Transcendentals

Question

Verify the formulas in Exercises by differentiation. $$\int(3 x+5)^{-2} d x=-\frac{(3 x+5)^{-1}}{3}+C$$

Step-by-Step Solution

Answer

The integral formula is verified by differentiation.

Step 1: Understand the Given Formula

We are tasked with verifying the integral formula \( \int (3x+5)^{-2} \, dx = -\frac{(3x+5)^{-1}}{3} + C \) by differentiating the right-hand side expression to check if it gives the original integrand, \((3x+5)^{-2}\).

Step 2: Differentiate the Right-hand Side

Differentiate the expression given on the right-hand side of the integral formula: \(-\frac{(3x+5)^{-1}}{3} + C\). This can be rewritten as \(-\frac{1}{3} (3x+5)^{-1} + C\). Differentiate using the chain rule: \(-\frac{1}{3} \cdot (-1)(3x+5)^{-2} \cdot 3\).

Step 3: Simplify the Result

Calculate the derivative: \(-\frac{1}{3} \times (-1) \times 3 \times (3x+5)^{-2} = (3x+5)^{-2}\). The constants \(-\frac{1}{3}\) and 3 cancel each other to leave \((3x+5)^{-2}\).

Step 4: Verify Correctness

Since the derivative \((3x+5)^{-2}\) matches the original integrand, \((3x+5)^{-2}\), this confirms that the given formula is correctly derived. Thus, the integration formula is verified.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. A derivative represents the rate of change of a function with respect to one of its variables. For the function given in the exercise, you will need to work with expressions that involve powers and more complex terms, like \( (3x + 5)^{-1} \). The process of differentiation lets us determine how changes in \( x \) affect the value of the function overall. By finding the derivative, you can identify slopes of curves at specific points, and this can show you how the function behaves as \( x \) moves. This is crucial for understanding the behavior of the function and is what allows us to verify the correct integral formula in the exercise. When differentiating \(-\frac{(3x+5)^{-1}}{3} + C\), the constant \( C \) disappears because the derivative of any constant is zero. This emphasizes that differentiation focuses on changes rather than fixed values. The original expression is transformed, showing its change in relation to \( x \). This ability to transition seamlessly from integrals to derivatives is a powerful tool in calculus.

Chain Rule

The chain rule is an essential differentiation technique used when dealing with composite functions. Composite functions involve one function nested inside another, like \( (3x+5)^{-1} \) in the exercise. To differentiate such functions, the chain rule provides a method: it involves taking the derivative of the outer function while simultaneously considering the derivative of the inner function. For instance, in our exercise, the outer function is essentially a power function, and the inner function is \( 3x+5 \).

• First, differentiate the outer function with respect to its inner part;
• Then multiply this by the derivative of the inner function.

Applied here, you take \(-\frac{1}{3} (3x+5)^{-1} + C\) as the outer function where power rule states that the derivative of \( x^n \) is \( n*x^{n-1} \), so you have \(-1(3x+5)^{-2}\) for the outer part. The derivative of the inner function \( 3x + 5 \) is 3. Multiply these results and combine the constants to ensure you get back to the original integrand. This process effectively "unwraps" the nested structure of functions so you can evaluate their behavior and verify formulas for accuracy.

Functions

Functions are the backbone of calculus, allowing us to describe mathematical relationships between variables. In this exercise, we are dealing with a function expressed as \( (3x+5)^{-2} \). Functions can describe a wide range of phenomena, from simple linear relationships to more complex and nonlinear ones.

• The basic idea is that you input a value (the input variable, usually \( x \)), and the function transforms it into another value (the output).
• Understanding how functions work is crucial when it comes to performing operations like differentiation and verifying integrals.

Here, the expression \( (3x+5)^{-2} \) is a power function with a shifted base, formed by an inner linear function, \( 3x + 5 \). This creates a specific type of curve when plotted on a graph, reflecting how the function manipulates its inputs. Visualization of functions like this can help clarify what happens when you apply differentiation or integration. Grasping these concepts allows you to appreciate how integrals and derivatives serve as tools to understand and solve real-world problems, by relating variation and accumulation concepts to a range of mathematical scenarios.