Indefinite Integral of Sinusoidal Powers

Find the indefinite integral of $\sin^5(x) \cdot \cos^2(x) \, dx$.

When approaching the integration of sinusoidal functions, particularly those raised to a power and multiplied together, a keen understanding of trigonometric identities is crucial. The key strategy in solving integrals of this nature involves expressing the product of trigonometric functions in a form that is more easily integrable. In some instances, this might involve using identities such as the Pythagorean identity, to simplify the expression to a form where basic integration techniques apply. Alternatively, you might consider a substitution method to reduce the complexity of the integral. In doing so, one often expresses one of the trigonometric functions in terms of another.

For this problem, where you have sine raised to the fifth power and cosine squared, a common approach is to leverage the identity that relates sine square to cosine square. This allows you to express higher powers of sine in terms of cosine and vice versa. Through substitution, you start by expressing powers of sine or cosine using double angle identities or expressing one function in another term. This technique not only simplifies the polynomial integrable expressions but also helps in navigating integrals that form when simplifying trigonometric products. By meticulously choosing these substitutions, the integrals become more straightforward, enabling integration via simpler antiderivatives, eventually allowing you to add the constant of integration, which characterizes indefinite integrals.

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