Problem 12 - Thomas Calculus

Question

Evaluate each integral in Exercises \(1-36\) by using a substitution to reduce it to standard form. $$ \int \frac{\cot (3+\ln x)}{x} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(\ln |\sin (3 + \ln x)| + C\).

Step 1: Identify a substitution

To simplify the integral, we need to perform a substitution. Notice that the term inside the cotangent function, \(3 + \ln x\), can be substituted to simplify the integral. Let's set \(u = 3 + \ln x\).

Step 2: Differentiate to find du

Differentiate \(u = 3 + \ln x\) with respect to \(x\). The derivative of \(3 + \ln x\) is \(\frac{d}{dx}(3) + \frac{d}{dx}(\ln x) = 0 + \frac{1}{x} = \frac{1}{x}\). Therefore, \(du = \frac{1}{x} dx\).

Step 3: Substitute in the integral

Rewrite the integral \(\int \frac{\cot (3+\ln x)}{x} \, dx\) using the substitution \(u = 3 + \ln x\) and \(du = \frac{1}{x} dx\). It becomes \(\int \cot u \, du\).

Step 4: Evaluate the integral

The integral \(\int \cot u \, du\) is a standard integral. The antiderivative of \(\cot u\) is \(\ln |\sin u| + C\), where \(C\) is the constant of integration.

Step 5: Back-substitute for x

Replace \(u\) with the original expression \(3 + \ln x\) to express the antiderivative in terms of \(x\). Thus, the solution is \(\ln |\sin (3 + \ln x)| + C\).

Key Concepts

Substitution Method

The substitution method is a powerful technique in integral calculus used to simplify integrals. The idea is to replace a portion of the integral with a single variable to simplify the integration process.
Essentially, it transforms a complex integral into a simpler one.
To apply the substitution method, follow these steps:

• Identify a part of the integral that can be replaced with a new variable.
• Express the new variable in terms of the original variable.
• Find the differential of this new variable to substitute it into the integral.

In the provided example, we identified the expression inside the cotangent function, \(3 + \ln x\), as a good candidate for substitution.
Setting \(u = 3 + \ln x\) allows simpler integration by converting the integral into a standard form, which is crucial for tackling more complex problems with ease.

Antiderivative

An antiderivative, also known as an indefinite integral, is a function whose derivative is the original function given in the integral.
In essence, finding the antiderivative is the reverse process of differentiation.
When you solve an integral, you are looking for a function whose rate of change (derivative) matches the function inside the integral (symbol). Finding the antiderivative involves determining that original function.

• Use known antiderivatives from calculus tables where possible.
• Apply integration techniques like substitution to find antiderivatives for complex functions.

For instance, in our example, once the substitution \(u = 3 + \ln x\) was made, the integral \(\int \cot u \, du\) needed an antiderivative.
We used the known result: \(\int \cot u \, du = \ln |\sin u| + C\), thus providing the solution in terms of \(u\).

Cotangent Function

The cotangent function, denoted as \(\cot x\), is a trigonometric function that represents the reciprocal of the tangent function.
In other words, \(\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}\).
It takes one real number input and gives another real number as output.Understanding how to integrate functions that include \(\cot x\) is essential.

• It often appears in calculus and advanced mathematics.
• Requires understanding trigonometric identities to simplify expressions.
• Often involves substitution for easier integration.

In the given problem, the cotangent function appears as part of the integral \(\int \frac{\cot (3+\ln x)}{x} dx\).
The function \(\cot u\) is integrated to \(\ln |\sin u| + C\) after substitution.

Logarithmic Differentiation

Logarithmic differentiation is a method used to differentiate complex functions, particularly useful when the function is a product of several expressions or contains an exponent.
It involves taking the natural logarithm of both sides of an equation and then differentiating.
This method simplifies the differentiation process by turning multiplication into addition (using log properties) and exponents into multiplication.

• Take the natural logarithm of both sides of a function \( y = f(x) \).
• Use properties of logarithms to simplify.
• Differentiate implicitly with respect to \(x\).

For example, with \(u = 3 + \ln x\), we use the derivative of \(\ln x\), which is \(\frac{1}{x}\).
This straightforward differentiation simplifies the transformation needed for substitution, making the integration process smoother and more approachable.