Integral of \( \sin^4(x) \)

Derivation and Evaluation

Evaluate the integral:

\[ \int \sin^4(x) \, dx \]

Rewrite as:

\[ = \int \sin^2(x) \sin^2(x) \, dx \]

Use the trigonometric identity \( \sin^2 x = \dfrac{1}{2}(1 - \cos(2x)) \) to rewrite the integral:

\[ = \dfrac{1}{4} \int (1 - \cos(2x))(1 - \cos(2x)) \, dx = \dfrac{1}{4} \int (1 - \cos(2x))^2 \, dx \]

Expand the integrand:

\[ = \dfrac{1}{4} \int (1 - 2\cos(2x) + \cos^2(2x)) \, dx \]

Use the trigonometric identity \( \cos^2 \theta = \dfrac{1}{2}(1 + \cos(2\theta)) \), which for \( \theta = 2x \) gives \( \cos^2(2x) = \dfrac{1}{2}(1 + \cos(4x)) \):

\[ = \dfrac{1}{4} \int \left( 1 - 2\cos(2x) + \dfrac{1}{2}(1 + \cos(4x)) \right) dx \]

Simplify terms inside the integral:

\[ = \dfrac{1}{4} \int \left( \dfrac{3}{2} - 2\cos(2x) + \dfrac{1}{2}\cos(4x) \right) dx \]

Use standard integrals like \( \int \cos(kx) \, dx = \dfrac{1}{k}\sin(kx) + c \) to evaluate the terms: