Problem 71 - Calculus for Biology and Medicine

Question

Compute the indefinite integrals. $$ \int \frac{\cos x}{1-\cos ^{2} x} d x $$

Step-by-Step Solution

Answer

The indefinite integral is \( -\csc x + C \).

Step 1: Identify the Trigonometric Identity

Recognize that the denominator can be simplified using the Pythagorean identity: \( 1 - \cos^2 x = \sin^2 x \). Therefore, the integral becomes \( \int \frac{\cos x}{\sin^2 x} \, dx \).

Step 2: Simplify the Integrand

Rewrite the integrand \( \frac{\cos x}{\sin^2 x} \) as \( \frac{1}{\sin^2 x} \cdot \cos x = \cot x \csc x \).

Step 3: Integrate Using Known Formula

Recall the known integral \( \int \cot x \csc x \, dx = -\csc x + C \). Apply this formula to compute \( \int \cot x \csc x \, dx \).

Step 4: Write the Final Answer

The integral evaluates to \( -\csc x + C \), where \( C \) is the constant of integration.

Key Concepts

Trigonometric Identity

Trigonometric identities are essential tools in simplifying and solving integrals involving trigonometric functions. In the given exercise, recognizing and applying a trigonometric identity is the crucial first step. We encounter an expression in the denominator: \( 1 - \cos^2 x \). By applying the Pythagorean identity, \( 1 - \cos^2 x = \sin^2 x \), we can transform this expression.

The Pythagorean identities are fundamental in trigonometry, providing relationships between the sine and cosine of an angle:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 - \cos^2 x = \sin^2 x \)
• \( 1 - \sin^2 x = \cos^2 x \)

By rewriting \( 1-\cos^2 x \) as \( \sin^2 x \), we successfully transform the original integral into a more workable form. Recognizing these identities aids us in simplifying the integrand significantly, making future steps easier to handle.

Simplifying Integrands

Simplifying the integrand is a helpful technique that turns complicated expressions into simpler forms, making integration more feasible. In this exercise, after using the trigonometric identity, the integrand becomes \( \frac{\cos x}{\sin^2 x} \). To simplify further, we separate the fraction into a product of two known trigonometric identities.

Let's see how this is done:

• Express \( \frac{\cos x}{\sin^2 x} \) as \( \frac{1}{\sin^2 x} \cdot \cos x \)
• We recognize \( \frac{1}{\sin^2 x} \) as \( \csc^2 x \) and \( \cos x \) as part of \( \cot x \cdot \csc x \)
• Thus, the integrand simplifies to \( \cot x \csc x \).

By rewriting the expression in terms of simple trigonometric functions like \( \cot x \) and \( \csc x \), we prepare the integral for straightforward application of known formulas.

Integral Formulas

Integral formulas are like shortcuts that allow us to solve integrals quickly without recalculating everything from scratch. In our example, once we simplify the integrand to \( \cot x \csc x \), we can use a readily available integral formula to find the solution.

This formula is based on standard integrals:

• \( \int \cot x \csc x \, dx = -\csc x + C \)

Here, \( C \) is a constant of integration that appears because we are dealing with indefinite integrals. The formula tells us the antiderivative of \( \cot x \csc x \) is \( -\csc x \), which we can now write as the final result, \( -\csc x + C \). Using these integral formulas saves us time and effort, allowing us to quickly integrate common trigonometric combinations.