Problem 58 - Thomas Calculus in SI Units

Question

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

Step-by-Step Solution

Answer

The derivative confirms the solution, verifying the formula is correct.

Step 1: Understand the Problem

We are given an integral and a proposed solution. We need to verify the correctness of the solution by differentiating it and checking if we arrive at the original integrand.

Step 2: Differentiate the Proposed Solution

Differentiate the given solution \[-\frac{(3x+5)^{-1}}{3} + C\]Using the chain and power rules, calculate the derivative.

Step 3: Apply the Chain Rule

To differentiate \(-(3x+5)^{-1}/3\), use the chain rule. Let \(u = 3x + 5\); hence, it becomes \[\frac{d}{dx} \left( -\frac{1}{3u} \right) = \frac{-1}{3} \cdot \frac{d}{du}(u^{-1}) \cdot \frac{du}{dx}\]This simplifies to \[\frac{-1}{3} \cdot (-1)u^{-2} \cdot 3\].

Step 4: Simplify the Expression

Substitute back to find the expression, \[-1 \cdot (3x+5)^{-2}\],which gives us the derivative of the proposed solution.

Step 5: Compare with Original Integrand

The derived expression \[-(3x+5)^{-2}\]is indeed the same as the original integrand, confirming that the solution is correct.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the derivative of a function, which is essentially the rate at which the function's output changes with respect to changes in the input. Think of it as how quickly a car accelerates or decelerates at a given moment. To differentiate a function, you need to know certain rules and techniques. This process involves:

• Determining the function you need to differentiate.
• Applying the appropriate rules, like the power rule or chain rule, to find the derivative.

In our specific problem, using differentiation helps us verify if the given integral solution is correct by checking if the differentiated result matches the original integrand.

Chain Rule

The chain rule is a handy tool in differentiation when dealing with composite functions. Composite functions are like a series of steps you take to reach a final result. For example, if you have a function within a function, you use the chain rule.To apply the chain rule, follow these steps:

• Identify the outer function and the inner function. In our case, the outer function is \(-u^{-1}\), and the inner function is \(3x + 5\).
• Differentiate the outer function with respect to the inner function.
• Then, differentiate the inner function with respect to the variable, which is \(x\) here.

Multiply these derivatives together to get the final result. This rule is essential for tackling complex equations in calculus and plays a key role in the differentiation process of the given solution.

Integral Calculus

Integral calculus, the counterpart to differential calculus, deals with the concept of integration. Integration is like adding up tiny pieces to find the whole. It can be thought of as finding the area under a curve.In the problem at hand, the goal is to verify an integration problem by checking the result through differentiation. Here’s what you typically do in integration:

• Recognize the function that needs to be integrated.
• Use integration rules to find the antiderivative, which is what reverses the process of differentiation.
• Include an integration constant \(C\), as integrals can have multiple valid constants attached.

In our exercise, by taking the derivative, we're ensuring that the `proposed solution was indeed a correct integral, thus verifying the accuracy of the integral calculus solution.

Power Rule

The power rule simplifies differentiation when dealing with power functions, where a variable is raised to a constant exponent. The rule states: for a function \(f(x) = ax^n\), the derivative is \(f'(x) = anx^{n-1}\).Applying the power rule is straightforward:

• Bring the exponent down as a coefficient.
• Subtract one from the original exponent.

In our exercise, the power rule is also utilized within the chain rule's application. Since we are differentiating \(u^{-1}\) (where \(u = 3x + 5\)), you:

• Differentiate \(u^{-1}\) using the power rule to get \(-u^{-2}\).
• Then finish applying the chain rule by differentiating \(3x + 5\).

This shows how the power rule fits within the broader process of differentiation. By mastering these rules, you can tackle complex calculus problems with confidence.