Problem 22 - Calculus of a Single Variable

Question

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{1}{x \ln x^{3}} d x $$

Step-by-Step Solution

Answer

The indefinite integral \( \int \frac{1}{x \ln x^3} dx =\frac{1}{3} \ln |\ln x^3| + C \)

Step 1: Determine the appropriate substitution

We can identify that the inverse of derivative of \( \ln x^3 \) is present in the integrand, so we will make a substitution \( u = \ln x^3 \). This is an appropriate substitution since it will simplify the given integrand.

Step 2: Determine du

The derivative of \( u = \ln x^3 \) with respect to \( x \) is \( \frac{3}{x} \), by the chain rule. Therefore, the differential \( du = \frac{3}{x} dx \). We can resolve \( dx \) from here as \( dx = \frac{du}{3/x} = \frac{x}{3}du \).

Step 3: Rewrite the integral

Now we will replace \( dx \) and \( \ln x^3 \) in our integral. We have \( \int \frac{1}{x \ln x^3} dx = \int \frac{1}{x u} \cdot \frac{x}{3}du = \frac{1}{3} \int \frac{1}{u} du \)

Step 4: Evaluate the integral

The integral \( \int \frac{1}{u} du \) is a simple log integral and can be directly integrated to \( \ln |u| \). Therefore, \( \frac{1}{3} \int \frac{1}{u} du = \frac{1}{3} \ln |u| + C \), where \( C \) is the constant of integration.

Step 5: Substitute u back into the integral

Now, we substitute \( u = \ln x^3 \) back into the integral. So, the final solution is \( \frac{1}{3} \ln |\ln x^3| + C \).

Key Concepts

Integration by Substitution

Integration by substitution, sometimes known as u-substitution, is a technique that involves changing the variable of integration to simplify the integral. It is particularly useful when dealing with composite functions or when the integral contains a function and its derivative.

To apply integration by substitution, you typically follow these steps:

• Identify a part of the integral that can be replaced with a new variable (u).
• Express this part as a function of x, that is, let u be a function of x.
• Take the derivative of u with respect to x to find 'du'.
• Rewrite the integral in terms of u and du by substituting the identified parts.
• Perform the integration with respect to u.
• Substitute back the expression of u in terms of x to get the result in the original variable.

Using integration by substitution can turn a difficult problem into a simpler one, much like finding a familiar pattern in an unfamiliar context.

For example, you are asked to integrate \(\frac{1}{x ewline \text{ln} x^{3}} dx\). Upon inspection, you notice that the derivative of \(\text{ln} x^{3}\) appears in the denominator multiplied by x, which suggests u-substitution. After setting \(u = \text{ln} x^3\) and finding the corresponding 'du', the integral becomes more manageable.

Natural Logarithm

The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.71828. The natural logarithm of a number x is commonly written as \(\text{ln} x\).

Properties of the natural logarithm include:

• \(\text{ln} (xy) = \text{ln} x + \text{ln} y\)
• \(\text{ln} (x^y) = y \text{ln} x\)
• It is undefined for non-positive values of x.

The natural logarithm plays a crucial role in various branches of mathematics including calculus, particularly in integration.

When faced with an integral involving a natural logarithm, such as \(\frac{1}{x ewline \text{ln} x^{3}} dx\), recognizing that you can manipulate and utilize logarithmic properties simplifies the problem. For example, the expression \(\text{ln} x^3\) can be transformed using logarithm properties to \(3 \text{ln} x\), which may assist in your strategy for integration.

Chain Rule

The chain rule is a fundamental derivative rule in calculus used to find the derivative of the composition of two or more functions. It states that if you have a function \(h(x) = f(g(x))\), then the derivative of h with respect to x is \(h'(x) = f'(g(x))g'(x)\).

In simpler terms, the chain rule allows you to differentiate a 'nested' function by multiplying the derivative of the outer function by the derivative of the inner function.

When applied to integration, especially in integration by substitution, the chain rule is used in reverse to identify the substitution that should be made. By recognizing a function and its derivative within the integral, you apply your understanding of the chain rule to make an effective substitution.

For the given integral \(\frac{1}{x ewline \text{ln} x^{3}} dx\), the derivative of \(\text{ln} x^3\) is necessary to determine 'du'. By applying the chain rule, you get \(\frac{3}{x}\) as the derivative. From this, it is clear that the derivative of the function inside the logarithm plays a significant role in the substitution process for integration.