Problem 59 - Calculus for Biology and Medicine

Question

Use the fact that $$ \cot x=\frac{\cos x}{\sin x} $$ to evaluate $$ \int \cot x d x $$

Step-by-Step Solution

Answer

The integral \( \int \cot x \, dx = \ln |\sin x| + C \).

Step 1: Understand the Integral to Solve

We are asked to evaluate the integral \( \int \cot x \, dx \). The given relation is \( \cot x = \frac{\cos x}{\sin x} \). We will use this expression to simplify our integral.

Step 2: Substitute Expression for \( \cot x \)

Substitute \( \cot x = \frac{\cos x}{\sin x} \) into the integral. This gives us \( \int \frac{\cos x}{\sin x} \, dx \). This substitution helps us recognize that the function can be integrated using a substitution method.

Step 3: Choose a Suitable Substitution

Let \( u = \sin x \). Then the derivative \( du = \cos x \, dx \). This will transform the integral into a simpler form with respect to \( u \).

Step 4: Perform the Substitution

Using the substitution \( u = \sin x \) and \( du = \cos x \, dx \), the integral becomes \( \int \frac{1}{u} \, du \). This is a standard integral.

Step 5: Integrate with Respect to \( u \)

The integral \( \int \frac{1}{u} \, du \) evaluates to \( \ln |u| + C \), where \( C \) is the constant of integration.

Step 6: Substitute Back to Original Variable

Replace \( u \) with \( \sin x \) in the integrated expression. Thus, \( \int \cot x \, dx = \ln |\sin x| + C \).

Key Concepts

Trigonometric Integrals

Trigonometric integrals are integral calculus problems involving trigonometric functions such as sine, cosine, and tangent. In this exercise, we are focusing on the integral of the cotangent function, which can be expressed as the ratio of cosine to sine: \( \cot x = \frac{\cos x}{\sin x} \). By rewriting cotangent in this form, it becomes easier to integrate.To evaluate these types of integrals, it is often helpful to rewrite the trigonometric function using identities. These identities simplify the given function and make it more recognizable for integration techniques, such as the substitution method. Breaking down the trigonometric function into familiar parts, like \( \frac{\cos x}{\sin x} \), helps in identifying a path to solve the integral effectively.

Substitution Method

The substitution method is a widely used technique in calculus to simplify complex integrals by transforming variables. This method involves changing a variable to rewrite the integral in a simpler form. In our exercise, we take \( u = \sin x \), which simplifies \( \int \frac{\cos x}{\sin x} \, dx \) into a form that is easier to solve.Here’s how it works:

• Identify a part of the integral that can be set as a new variable \( u \).
• For our integral, \( u \) was chosen to be \( \sin x \), making it convenient because the derivative \( du = \cos x \, dx \) was also present.
• Substitute these into the integral: \( \int \frac{1}{u} \, du \).

The substitution method turns a challenging trigonometric integral into a recognizable and easier-to-solve form, paving the way for further integration.

Antiderivatives

Antiderivatives are functions that reverse the process of differentiation, essentially finding the original function before it was differentiated. In this exercise, after using the substitution method, we arrive at an integral of the form \( \int \frac{1}{u} \, du \). This is known to have the antiderivative \( \ln |u| + C \), where \( C \) represents the constant of integration.Why is this important? Recognizing standard antiderivatives allows us to easily find solutions to integrals once they are transformed into simpler expressions. By knowing that \( \int \frac{1}{u} \, du = \ln |u| + C \), and substituting back \( u = \sin x \), we find that:

• The antiderivative of \( \int \cot x \, dx \) becomes \( \ln |\sin x| + C \).

A good grasp of antiderivatives provides the tools needed to solve definite and indefinite integrals efficiently, giving insight into the area under curves and the underlying functions.