Problem 347 - Calculus Volume 1

Question

For the following exercises, verify the derivatives and antiderivatives. $$\int \frac{d x}{x \ln (x) \ln (\ln x)}=\ln (\ln (\ln x))+C$$

Step-by-Step Solution

Answer

Yes, the antiderivative is correct upon verification.

Step 1: Differentiate the Right Side

We start by taking the derivative of the right side, which is \(\ln(\ln(\ln x)) + C\). Since \(C\) is a constant, it becomes zero upon differentiation. We need to use the chain rule for the derivative of \(\ln(\ln(\ln x))\).

Step 2: Apply the Chain Rule

Applying the chain rule, the derivative of \(\ln(\ln(\ln x))\) is \(\frac{1}{\ln(\ln x)} \cdot \frac{1}{\ln x} \cdot \frac{1}{x}\). This simplifies to \(\frac{1}{x \ln x \ln(\ln x)}\).

Step 3: Compare with the Original Integrand

The original integrand was \(\frac{1}{x \ln x \ln(\ln x)}\). This is the same as what we have obtained from differentiating the antiderivative. Thus, the antiderivative given in the exercise is verified to be correct.

Key Concepts

Derivative verification

When verifying derivatives, it's crucial to check that the derivative of the proposed antiderivative matches the original integrand. This step ensures that both integration and differentiation processes have been handled correctly.

To verify derivatives, you often differentiate the function that results from integrating your original expression. If you end up with your starting function (the integrand), then the antiderivative is confirmed. This is a fundamental check in calculus that reassures you that the original equation was solved properly.

Antiderivative

An antiderivative is essentially a function whose derivative is the given function. In calculus, finding an antiderivative allows us to obtain a broader understanding of the area under the curve represented by the original function.

For example, in our exercise, \(\ln(\ln(\ln x)) + C\) is the antiderivative of \((\frac{1}{x \, \ln x \, \ln(\ln x)})\). Antiderivatives always include '+ C' at the end, representing the constant of integration. This constant is crucial because it accounts for all possible vertical shifts of the curve, reflecting the family of functions that could yield the same derivative.

Chain rule

The chain rule is an essential concept in calculus used to differentiate compositions of functions. It is applied when you have a function inside of another function, and it allows you to take the derivative of the whole expression efficiently.

In our exercise, the chain rule is used to find the derivative of \(\ln(\ln(\ln x))\). It requires taking successive derivatives of each nested function layer. In practical terms, it helps simplify complex functions by breaking them down into single-variable functions whose derivatives are easier to compute.

• First, take the derivative of the outer function, keeping the inner functions intact.
• Then, multiply by the derivative of the inner functions as you proceed inward.

This rule streamlines the differentiation of intricate compositions and ensures calculations proceed correctly.

Integral calculus

Integral calculus focuses on finding the total size or value, such as area under a curve. It provides tools to move backwards from derivative functions to find original functions. Two main concepts involved are indefinite integrals (antiderivatives) and definite integrals (finding areas under curves between specific bounds).

The given exercise involves evaluating an indefinite integral, where resolving the antiderivative shows the general form of the integral's solution.

Working through integral calculus enhances understanding of mathematical phenomena involving accumulation and area, making it a cornerstone of many fields in science and engineering.