How do you find the integral of # Cos(2x)Sin(x)dx#?

Answer 1

#=cosx - 2/3cos^3x + C#

Use the identity #cos(2x) = 1 - 2sin^2x#.
#=int(1 - 2sin^2x)sinxdx#

Multiply out.

#=int(sinx - 2sin^3x)dx#
Separate using #int(a + b)dx = intadx + intbdx#
#=int(sinx)dx - int(2sin^3x)dx#
The antiderivative of #sinx# is #-cosx#. Use the property of integrals that #int(Cf(x))dx = Cintf(x)# where #C# is a constant. Note that #sin^3x# can be factored as #sin^2x(sinx)#, which can in turn be written as #(1- cos^2x)(sinx)# by the identity #sin^2x + cos^2x = 1#.
#=-cosx - 2int(1 - cos^2x)sinxdx#
Let #u = cosx#. Then #du = -sinxdx -> dx = -(du)/sinx#.
#=-cosx - 2int(1 - u^2)sinx * -(du)/sinx#

The sines under the integral cancel each other out.

#=-cosx - 2int(1 - u^2) * -(du)#
Extract the negative #1#.
#=-cosx + 2int(1 - u^2)du#

Separate the integrals.

#=-cosx + 2int1du - 2intu^2du#
Integrate using the rule #int(x^n)dx = x^(n + 1)/(n + 1) + C#, where #C# is a constant.
#=-cosx + 2u - 2(1/3u^(3)) + C#
Resubstitute #u = cosx# to define the function with respect to #x#.
#=-cosx + 2cosx - 2/3cos^3x + C#

Finally, combine like terms.

#=cosx - 2/3cos^3x + C#

Hopefully this helps!

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Answer 2

To find the integral of ( \cos(2x)\sin(x) ), you can use integration by parts method. Let ( u = \sin(x) ) and ( dv = \cos(2x)dx ). Then, ( du = \cos(x)dx ) and ( v = \frac{1}{2}\sin(2x) ).

Applying the integration by parts formula:

[ \int u dv = uv - \int v du ]

Substituting the values:

[ \int \cos(2x)\sin(x)dx = \frac{1}{2}\sin(2x)\sin(x) - \int \frac{1}{2}\sin(2x)\cos(x)dx ]

Now, you can integrate the remaining integral using the same method, or through trigonometric identities.

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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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