Integral of \( \cos^3 x \)

Derivation and Evaluation

Find the integral:

\[ \int \cos^3 x \, dx \]

Write the integral by splitting off a single cosine factor:

\[ \int \cos^3 x \, dx = \int \cos^2 x \cos x \, dx \]

Use the trigonometric identity \( \cos^2 x = 1 - \sin^2 x \) to rewrite the integrand:

\[ \int \cos^3 x \, dx = \int (1 - \sin^2 x) \cos x \, dx \]

Expand the integrand and split the integral into two parts:

\[ \int \cos^3 x \, dx = \int \cos x \, dx - \int \sin^2 x \cos x \, dx \]

Use Integration by Substitution for the second term: let \( u = \sin x \), which gives \( \dfrac{du}{dx} = \cos x \) or \( du = \cos x \, dx \). Substituting this yields:

\[ \int \cos^3 x \, dx = \int \cos x \, dx - \int u^2 \, du \]

Use standard integral formulas to evaluate each integral:

\[ \int \cos^3 x \, dx = \sin x - \dfrac{1}{3} u^3 + c \]

where \( c \) is the constant of integration.

Substitute back \( u = \sin x \) to find the final result: