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

Evaluate the integral \[ \int \sin^4(x) \; dx \] Rewrite as \[ = \int \sin^2(x) \sin^2(x) \; dx \]
Use the trigonometric identity \( \sin^2 x = \dfrac{1}{2} (1 - \cos (2 x) ) \) to rewrite as
\[ = \dfrac{1}{4} \int (1 - \cos (2 x) ) (1 - \cos (2 x) ) \; dx \]
Expand the the integrand \[ = \dfrac{1}{4} \int ( 1 - 2 \cos (2) + \cos^2 (2 x) ) \; dx \] Use trigonometric identity \( \cos^2 x = \dfrac{1}{2} (1 + \cos (2 x) ) \) to change \( \cos^2 (2 x) \) included in the integrand \[ = \dfrac{1}{4} \int ( 1 - 2 \cos (2 x) + \dfrac{1}{2} (1 + \cos (4 x) ) ) \; dx \] Simplify \[ = \dfrac{1}{4} \int ( \dfrac{3}{2} - 2 \cos (2 x) + \dfrac{1}{2} \cos (4 x) ) \; dx \] Use the common integral \( \int \cos (kx) \; dx = \dfrac{1}{k} \sin (kx) + c \) to evaluate the given integral \[ \boxed {\int \sin^4(x) \; dx = \dfrac{3}{8} x - \dfrac{1}{4} \sin (2x) + \dfrac{1}{32} \sin (4x) + c} \]

More References and Links

1. Table of Integral Formulas
2. University Calculus - Early Transcendental - Joel Hass, Maurice D. Weir, George B. Thomas, Jr., Christopher Heil - ISBN-13 : 978-0134995540
   
3. Calculus - Gilbert Strang - MIT - ISBN-13 : 978-0961408824
   
4. Calculus - Early Transcendental - James Stewart - ISBN-13: 978-0-495-01166-8