How do you integrate #sin^(3)2xdx#?

Answer 1
#int sin^3 (2x) dx = int (sin(2x))^3 dx = int (2sinx cosx)^3 dx#, So
#int sin^3 (2x) dx = 8int sin^3x cos^3x dx#
Now either pull off #sin^2x# and write the integrand as #(1-cos^2x)cos^3x sinx dx# (So we have powers of sine times differential of sine) or vice versa (details follow)
#8int sin^3x cos^3x dx = 8int sin^3 x (cos^2x) cosx dx# (Do you see where we're going?)
#=8int sin^3 x (1-sin^2x) cosx dx# See it yet?
#=8int (sin^3 x -sin^5x) cosx dx#
#=8int (sin^3 x) cosx dx -int (sin^5x) cosx dx#
#=8(1/4sin^4x - 1/6 sin^6x) +C#
#= 2sin^4x-4/3sin^6x+C#
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Answer 2
#int sin^3(2x)dx#

There's no obvious replacement and no obvious parts, so I can't integrate right away.

What do I know about Sine? #d/(dx)(sinx)=cosx# , #int sinx dx = -cosx#

That covers if in calculus. In trigonometry, there are additional

#sin^2(2x) + cos^2 (2x)x = 1#
so #sin^2 (2x)=1-cos^2(2x)# and I know the derivative of cos is -sin, so maybe a substitution after all. TRY IT. (I know it will work by many years of experience. A student just has to try something and see if it helps. (So do I, for higher level problems.)

Try it:

#int sin^3(2x)dx = int sin^2 (2x) sin(2x)dx#
#= int (1-cos^2 (2x)) sin (2x) dx = int [sin(2x) - cos^2 (2x) sin(2x)] dx#

I ought to be able to combine each of these two terms now:

#int sin(2x) dx = -1/2 cos (2x)#
and #int - cos^2 (2x) sin(2x) dx#, by letting #u=cos(2x)# we can get
#int - cos^2 (2x) sin(2x) dx = 1/6 cos^3 (2x)#

Thus, we obtain:

#-1/2 cos (2x)+1/6 cos^3 (2x) +C#

Or view the alternative solution I'll share.

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Answer 3

To integrate sin^(3)(2x) dx, you can use the trigonometric identity:

sin^3(2x) = (1/4)(3sin(2x) - sin(6x))

Then, integrate each term separately:

∫(1/4)(3sin(2x) - sin(6x)) dx

= (1/4) * (∫3sin(2x) dx - ∫sin(6x) dx)

= (1/4) * (-3/2cos(2x) + (1/6)cos(6x)) + C

= -(3/8)cos(2x) + (1/24)cos(6x) + C

where C is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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