Integral of \( \sin^3 x \)

Evaluate the integral \[ \int \sin^3 x \; dx \] Write the integrand \( \sin^3 x \) as a product of as \( \sin^2 x \) and \( \sin x \) \[ \int \sin^3 x \; dx = \int \sin^2 x \sin x \; dx\] Use of the trigonometric identity \( \; \sin^2 x = 1 - \cos^2 x \) to write the integral as
\[ \int \sin^3 x \; dx = \int (1 - \cos^2 x) \sin x \; dx\]
Expand \( (1 - \cos^2 x) \sin x \) and write \[ \int \sin^3 x \; dx = \int \sin x \; dx - \int \cos^2 x \sin x \; dx \] Integration by Substitution : Let \( u = \cos x \) and hence \( \dfrac{du}{dx} = - \sin x \) or \( du = - \sin x \; dx \) and substitute to obtain \[ \int \sin^3 x \; dx = \int \sin x \; dx + \int u^2 \; du \] Use integral formulas to evaluate the above integral and write \[ \int \sin^3 x \; dx = - \cos x + \dfrac{1}{3} u^3 + c \] Substitute back \( u = \cos x \) to find the final answer \[ \boxed { \int \sin^3 x \; dx = - \cos x + \dfrac{1}{3} \cos^3 x + c } \]

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