Integrate (cosx)^3

To find the integral of \( \cos^3(x) \), we can utilise a trigonometric identity to simplify the expression. The identity states that \( \cos^2(x) = 1 – \sin^2(x) \). By substituting this identity into our integral, we can express \( \cos^3(x) \) as \( \cos(x) \cdot \cos^2(x) \), which can be rewritten as \( \cos(x) \cdot (1 – \sin^2(x)) \). This allows us to separate the integral into two distinct parts, making it easier to evaluate.

Next, we can express the integral as the sum of two separate integrals: \( \int \cos(x) \, dx – \int \cos(x) \sin^2(x) \, dx \). The first integral, \( \int \cos(x) \, dx \), is straightforward and evaluates to \( \sin(x) + C \), where \( C \) is the constant of integration. The second integral, \( \int \cos(x) \sin^2(x) \, dx \), requires the use of substitution. By letting \( u = \sin(x) \), we find that \( du = \cos(x) \, dx \). This transforms the integral into \( \int u^2 \, du \), which can be easily integrated.

Upon integrating \( u^2 \), we obtain \( \frac{u^3}{3} + C \). Substituting back for \( u \), we have \( \frac{\sin^3(x)}{3} + C \). Therefore, combining both parts of the integral, we arrive at the final result for the integral of \( \cos^3(x) \): \( \int \cos^3(x) \, dx = \sin(x) – \frac{\sin^3(x)}{3} + C \). This expression provides a complete solution to the integral, incorporating both the trigonometric identity and the method of substitution effectively.