Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx

Verified step by step guidance

1. Rewrite the integrand to express it in terms of sine and cosine functions. Recall that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\), so the integral becomes \(\int \frac{\cot(x)}{\cos^{2}(x)} \, dx = \int \frac{\cos(x)}{\sin(x) \cos^{2}(x)} \, dx\).
2. Simplify the integrand by canceling one \(\cos(x)\) term in the numerator and denominator, resulting in \(\int \frac{1}{\sin(x) \cos(x)} \, dx\).
3. Consider a substitution to simplify the integral. One useful substitution is to let \(u = \sin(x)\), which implies \(du = \cos(x) \, dx\). Rearranging, we get \(dx = \frac{du}{\cos(x)}\).
4. Substitute \(u\) and \(dx\) back into the integral. The integral becomes \(\int \frac{1}{u \cos(x)} \cdot \frac{du}{\cos(x)} = \int \frac{1}{u \cos^{2}(x)} \, du\). Since this still contains \(\cos(x)\), consider an alternative substitution or rewrite the integrand differently.
5. Alternatively, rewrite the original integrand as \(\cot(x) \sec^{2}(x)\) because \(\frac{1}{\cos^{2}(x)} = \sec^{2}(x)\). Then, use the substitution \(u = \sin(x)\), so \(du = \cos(x) \, dx\). Express \(\cot(x) \sec^{2}(x) \, dx\) in terms of \(u\) and \(du\) to evaluate the integral.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. They help simplify integrals by rewriting expressions in more manageable forms, such as expressing cotangent as cos(x)/sin(x) or using Pythagorean identities to transform powers of sine and cosine.

Integration Techniques

Integration techniques include methods like substitution, integration by parts, and rewriting integrals to simpler forms. For integrals involving trigonometric functions, substitution is often used by identifying a function and its derivative within the integrand to simplify the integral into a basic form.

Handling Rational Trigonometric Functions

Rational trigonometric functions are ratios of trigonometric expressions, such as cot(x)/cos²(x). Understanding how to manipulate these ratios, often by expressing all terms in sine and cosine, is essential to simplify the integral and apply substitution or other integration methods effectively.