Integral of \( \sin^3(x) \cos^2(x) \)

Evaluate the integral \[ \int \sin^3(x) \cos^2(x) \; dx \] Rewrite as \[ = \int \sin^2(x) \cos^2(x) \sin x \; dx \]
Use trigonometric identity \( \sin^2 x = 1 - \cos^2 x \) and substitute
\[ = \int (1 - \cos^2 x) \cos^2(x) \sin x \; dx \]
Expand the the integrand \[ = \int (\cos^2(x) - \cos^4(x)) \sin x \; dx \] Use Integration by Substitution : \( u = cos x \) so that \( du = - \sin x \; dx \) \[ = - \int (u^2 - u^4) \; du \] Use the common integral \( \int u^n du = \dfrac{1}{n+1} u^{n+1}+ c\) to evaluate the above integral \[ - \dfrac{1}{3} u^3 + \dfrac{1}{5} u^5 + c \] Substitute back \( u = cos x \) to obtain the final answer \[ \boxed { \int \sin^3(x) \cos^2(x) \; dx = - \dfrac{1}{3} \cos^3 (x) + \dfrac{1}{5} \cos^5 (x) + c } \]

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