85. Another form of ∫ sec x dx
a. Verify the identity:
sec x = cos x / (1 - sin² x)
b. Use the identity in part (a) to verify that:
∫ sec x dx = (1/2) ln |(1 + sin x)/(1 - sin x)| + C

Verified step by step guidance

1. Step 1: Start by verifying the identity given in part (a): \( \sec x = \frac{\cos x}{1 - \sin^2 x} \). Recall the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), and use it to rewrite the denominator \( 1 - \sin^2 x \) as \( \cos^2 x \).
2. Step 2: Substitute \( 1 - \sin^2 x = \cos^2 x \) into the right-hand side of the identity to get \( \frac{\cos x}{\cos^2 x} = \frac{1}{\cos x} \), which is exactly \( \sec x \). This completes the verification of the identity in part (a).
3. Step 3: For part (b), start with the integral \( \int \sec x \, dx \) and use the identity from part (a) to rewrite \( \sec x \) as \( \frac{\cos x}{1 - \sin^2 x} \). This transforms the integral into \( \int \frac{\cos x}{1 - \sin^2 x} \, dx \).
4. Step 4: Use the substitution \( u = \sin x \), which implies \( du = \cos x \, dx \). This substitution simplifies the integral to \( \int \frac{1}{1 - u^2} \, du \).
5. Step 5: Recognize that \( \frac{1}{1 - u^2} \) can be decomposed using partial fractions into \( \frac{1}{(1 - u)(1 + u)} \). Integrate the resulting expression to obtain \( \frac{1}{2} \ln \left| \frac{1 + u}{1 - u} \right| + C \). Finally, substitute back \( u = \sin x \) to complete the verification.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. Understanding how to manipulate and verify identities, such as expressing sec x in terms of sine and cosine, is essential for simplifying integrals and proving equivalences.

Integration of Trigonometric Functions

Integrating trigonometric functions often requires substitution or using known identities to rewrite the integrand. Recognizing alternative forms of sec x and applying appropriate substitutions helps in evaluating integrals that are not straightforward.

Logarithmic Integration Results

Some integrals of trigonometric functions result in logarithmic expressions involving absolute values. Understanding how to derive and interpret these logarithmic forms, such as the integral of sec x leading to a natural logarithm expression, is crucial for verifying integral formulas.