0Let's solve an example using 'u' substitution method.
∫ sin² x cos x dx ------------------------A
As we can see " cos x " along with " sin x " and we don't have any ready made formula for the product of two functions in integration, let's try to eliminate one of them. As cos x is raised to power one, we can eliminate cos x by using ' u ' substitution.
Let u = sin x, by differentiating "u " wrt "x " du / dx = cos x or dx = du / cos x
By substituting "u" and "dx" in "A"
∫ sin² x cos x dx = ∫ u² cos x du / cos x
= ∫ u² cos x / cos x du. ( cos x cancels out )
= ∫ u² du
= u³ / 3 + c
=1/3 sin³ x + c. (by substituting "sin x" back in place of " u ")
∴ ∫ sin² x cos x dx = 1/3 sin³ x + c
