Problem 70 - 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 integral is \( -\csc x + C \).

Step 1: Simplify the Integrand

The expression for the integrand is \( \frac{\cos x}{1 - \cos^2 x} \). Notice that \( 1 - \cos^2 x = \sin^2 x \) (using the Pythagorean identity for sine and cosine). Thus, the integrand becomes \( \frac{\cos x}{\sin^2 x} \), which can be rewritten as \( \cos x \cdot \csc^2 x \).

Step 2: Rewrite the Expression for Integration

Rewrite the expression \( \frac{\cos x}{\sin^2 x} \) as \( \frac{1}{\sin x} \cdot \frac{\cos x}{\sin x} \), which is equivalent to \( \cot x \cdot \csc x \). This simplification gives us the integral \( \int \cot x \csc x \, dx \).

Step 3: Integrate the Simplified Form

We know that the derivative of \( \csc x \) is \( -\cot x \csc x \). Therefore, the integral \( \int \cot x \csc x \, dx \) can be solved by realizing it as the negative of the derivative: \( \int \cot x \csc x \, dx = -\csc x + C \), where \( C \) is the constant of integration.

Key Concepts

Trigonometric Identities

Trigonometric identities are incredibly useful tools in calculus, especially when dealing with integrals involving trigonometric functions. These identities help simplify complex expressions so they can be integrated more easily.

Some of the key trigonometric identities include:

• Pythagorean identities, which relate the squares of sine and cosine functions, such as \( \sin^2 x + \cos^2 x = 1 \).
• Reciprocal identities, which connect basic trigonometric functions to their reciprocals, like \( \csc x = \frac{1}{\sin x} \) and \( \sec x = \frac{1}{\cos x} \).
• Quotient identities, which express one trigonometric function in terms of the division of others, such as \( \tan x = \frac{\sin x}{\cos x} \).

In our exercise, knowing that \( 1 - \cos^2 x = \sin^2 x \) is crucial.

This Pythagorean identity lets us rewrite more complex functions into simpler ones, leading to valid rearrangements like transforming \( \frac{\cos x}{1 - \cos^2 x} \) into \( \frac{\cos x}{\sin^2 x} \). By making use of these identities, we can break down and solve integrals more effectively.

Pythagorean Identity

The Pythagorean identity is a fundamental trigonometric identity that relates the squares of the sine and cosine functions:\[ \sin^2 x + \cos^2 x = 1 \]

This identity is derived from the Pythagorean theorem and is true for all angles \( x \). By manipulating this identity, we can obtain other useful forms:

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

In the context of integrating the expression \( \frac{\cos x}{1 - \cos^2 x} \), the identity allows us to substitute \( 1 - \cos^2 x \) with \( \sin^2 x \).

This substitution is particularly helpful because it results in a more integrable form.

Realizing such transformations is essential as it not only simplifies integration but also highlights the interconnected nature of trigonometric functions.

Integration Techniques

When facing trigonometric integrals, a variety of techniques are available to make the integration process easier.

Here are a few common techniques that can aid in solving these types of integrals:

• **Substitution:** This approach involves substituting a part of the integrand with a single variable to simplify the integral. It is often used when recognizing a function and its derivative within the integral.
• **Trigonometric Identities:** As demonstrated in our exercise, employing identities like the Pythagorean identity helps transform complex expressions into simpler ones.
• **Integration by Parts:** This technique is useful when the integrand is a product of two functions. It exploits the product rule in reverse to break down the integral into more manageable parts.

In our example, after utilizing trigonometric identities, we converted the original integrand into a simpler form: \( \int \cot x \csc x \, dx \).

This form is easily integrable by recognizing that the derivative of \( \csc x \) is \( -\cot x \csc x \). Therefore, by recognizing derivatives of basic trigonometric functions, we can streamline the integration process and swiftly reach the result \( -\csc x + C \).

Integration techniques and the recognition of patterns greatly assist in handling more complex integral problems.