Integral of \( \sin^2 x \)

Derivation and Evaluation

Find the integral:

\[ \int \sin^2 x \, dx \]

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

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

Apply the sum rule of integrals \( \displaystyle \int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx \) to rewrite the expression:

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

Evaluate using standard integrals \( \displaystyle \int 1 \, dx = x \) and \( \displaystyle \int \cos(2x) \, dx = \dfrac{1}{2} \sin(2x) \):

\[ = \dfrac{1}{2} x - \dfrac{1}{2} \left( \dfrac{1}{2} \sin(2x) \right) + c \]

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

Simplifying the terms, the final result is given by: