# How do you find the antiderivative of #cos^4 (2x) * sin^3 (2x)#?

Here, let

Here, let

Therefore,

Therefore,

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To find the antiderivative of ( \cos^4(2x) \cdot \sin^3(2x) ), you can use the method of substitution. Let ( u = \cos(2x) ). Then, ( du = -2\sin(2x) , dx ).

Using this substitution, rewrite the integral in terms of ( u ) and ( du ):

[ \int \cos^4(2x) \cdot \sin^3(2x) , dx = \int u^4 \cdot \left(\frac{-du}{2}\right)^3 ]

Now, simplify and integrate the expression in terms of ( u ):

[ = -\frac{1}{8} \int u^4 , du ] [ = -\frac{1}{8} \cdot \frac{u^5}{5} + C ]

Finally, substitute back for ( u ):

[ = -\frac{1}{8} \cdot \frac{\cos^5(2x)}{5} + C ]

So, the antiderivative of ( \cos^4(2x) \cdot \sin^3(2x) ) is ( -\frac{1}{40} \cos^5(2x) + C ), where ( C ) is the constant of integration.

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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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