Antiderivative of Cosine to the Fifth Power

Find the anti-derivative of $\cos^5(x)$.

When tasked with finding the antiderivative of trigonometric expressions like cosine raised to a power, it's crucial to utilize trigonometric identities and integration strategies. An expression such as cosine to the fifth power can often be decomposed using identities that simplify odd powers of cosine, allowing us to express them in terms of lower powers which are easier to integrate. This often involves recognizing patterns that can be aligned with known integrals or applying substitutions that simplify the expression.

In this specific case, a useful strategy involves breaking the power of cosine into parts that allow the substitution of sine or cosine to work effectively with the integral. For instance, separating one cosine from the product, transforming to sine using the Pythagorean identity, and then determining an appropriate substitution can streamline the process. Recognizing opportunities to use substitution is a critical skill when working with trigonometric integrals, as it simplifies not only the computation but also the manipulation of the integral into more recognizable forms. This approach highlights the strategy of reducing complex integrals into simpler ones through clever transformations and substitutions, an essential technique for mastering integration.

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