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.

Question

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Respuesta :

okpalawalter8 okpalawalter8 · Answer 1 · 2020-09-06T04:12:52+02:00

Complete

Find the integral of [tex]f(x) = 16sin^2 (x ) cos(x) dx[/tex]

Answer:

The solution is [tex]\frac{16}{3} sin^{3}x + c[/tex]

Step-by-step explanation:

So

[tex]Let \ u = sin(x)[/tex]

=> [tex]\frac{du}{dx} = cos (x)[/tex]

=> [tex]du = cos(x)dx[/tex]

So

[tex]\int\limits {16sin^2 (x ) cos(x) dx} \, \equiv \int\limits {16u^2 du}[/tex]

=> [tex]\int\limits {16u^2 du} = 16 [\frac{u^3}{3} ] + c[/tex]

Now substituting sin(x) for u

[tex]\frac{16}{3} u^3 + c = \frac{16}{3} sin^{3}x + c[/tex]

So the integral of [tex]f(x) = 16sin^2 (x ) cos(x) dx[/tex] is

[tex]\frac{16}{3} sin^{3}x + c[/tex]