Question

Since cos(x) is the derivative of sin(x), then integral 16 sin^2(x) cos(x) dx can be done by substituting u = sin (x) sin (x) and du = cos (x) cos (x)^dx. With the substitution u = sin(x), we get integral 16 sin^2 (x) cos(x) dx = 16 integral u^2 du. which integrates to + C. Substituting back in to get the answer in terms of sin(x), we have integral 16 sin^2 (x) cos (x) dx = + C.

Answer 1

Complete

Find the integral of
`f(x) =  16sin^2 (x ) cos(x) dx`

Answer:

The solution is
`(16)/(3) sin^(3)x + c`

Explanation:

So

`Let \  u  =  sin(x)`

=>
`(du)/(dx)  =  cos (x)`

=>
`du  =  cos(x)dx`

So

`\int\limits {16sin^2 (x ) cos(x) dx} \, \equiv \int\limits {16u^2  du}`

=>
`\int\limits {16u^2  du} = 16 [(u^3)/(3) ] + c`

Now substituting sin(x) for u

`(16)/(3) u^3 + c  =  (16)/(3) sin^(3)x + c`

So the integral of
`f(x) =  16sin^2 (x ) cos(x) dx` is

`(16)/(3) sin^(3)x + c`