Use any method to evaluate the integrals in Exercises 65–70.
∫ sin³(x) / cos⁴(x) dx

Verified step by step guidance

1. Rewrite the integrand \( \frac{\sin^3(x)}{\cos^4(x)} \) by expressing \( \sin^3(x) \) as \( \sin(x) \cdot \sin^2(x) \). This gives \( \int \frac{\sin(x) \cdot \sin^2(x)}{\cos^4(x)} \, dx \).
2. Use the Pythagorean identity \( \sin^2(x) = 1 - \cos^2(x) \) to rewrite \( \sin^2(x) \) in terms of \( \cos(x) \). Substitute this into the integral to get \( \int \frac{\sin(x) (1 - \cos^2(x))}{\cos^4(x)} \, dx \).
3. Make the substitution \( u = \cos(x) \), which implies \( du = -\sin(x) \, dx \) or equivalently \( -du = \sin(x) \, dx \). Replace \( \sin(x) \, dx \) in the integral with \( -du \).
4. Rewrite the integral entirely in terms of \( u \): \( \int \frac{\sin(x)(1 - u^2)}{u^4} \, dx = - \int \frac{1 - u^2}{u^4} \, du \).
5. Split the integral into simpler terms: \( - \int \left( u^{-4} - u^{-2} \right) du \). Then integrate each term separately using the power rule for integrals.

Key Concepts

Trigonometric Identities and Manipulations

Understanding how to rewrite powers of sine and cosine using identities or algebraic manipulation is essential. For example, expressing sin³(x) as sin(x)·sin²(x) and then using sin²(x) = 1 - cos²(x) helps simplify the integral into a more manageable form.

Substitution Method

The substitution method involves changing variables to simplify the integral. In this case, substituting u = cos(x) transforms the integral into a rational function of u, making it easier to integrate by reducing trigonometric complexity.

Integration of Rational Functions

After substitution, the integral often becomes a rational function, which requires techniques like polynomial division or partial fraction decomposition. Mastery of these methods allows for straightforward integration of the resulting algebraic expression.