# How do you integrate #int sin^(4) (2x) #?

When dealing with even powers of sine and cosine, the cosine double-angle identity is a great tool to use if substitution is not an option.

Keep in mind that the following forms are comparable:

The cosine function's argument must contain twice as much as the sine function's. Next, we can rewrite the integrand using the blue identity:

Which is comparable to:

Changing this to:

which can subsequently be divided and combined as before:

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To integrate ( \int \sin^4(2x) ), you can use the power-reducing identity for sine, which states that ( \sin^2(x) = \frac{1 - \cos(2x)}{2} ).

So, we can rewrite ( \sin^4(2x) ) as ( (\sin^2(2x))^2 ).

Substituting ( \sin^2(2x) = \frac{1 - \cos(4x)}{2} ), we get:

[ (\sin^2(2x))^2 = \left(\frac{1 - \cos(4x)}{2}\right)^2 ]

Expand and simplify the expression, then integrate term by term.

[ \left(\frac{1 - \cos(4x)}{2}\right)^2 = \frac{1 - 2\cos(4x) + \cos^2(4x)}{4} ]

[ = \frac{1}{4} - \frac{1}{2}\cos(4x) + \frac{1}{4}\cos^2(4x) ]

To integrate ( \frac{1}{4} - \frac{1}{2}\cos(4x) + \frac{1}{4}\cos^2(4x) ), you integrate each term separately:

[ \int \frac{1}{4} , dx - \int \frac{1}{2}\cos(4x) , dx + \int \frac{1}{4}\cos^2(4x) , dx ]

[ = \frac{1}{4}x - \frac{1}{8}\sin(4x) + \frac{1}{4} \int (1 + \cos(8x)) , dx ]

[ = \frac{1}{4}x - \frac{1}{8}\sin(4x) + \frac{1}{4}\left(x + \frac{1}{8}\sin(8x)\right) + C ]

[ = \frac{1}{4}x + \frac{1}{4}x - \frac{1}{8}\sin(4x) + \frac{1}{64}\sin(8x) + C ]

[ = \frac{1}{2}x - \frac{1}{8}\sin(4x) + \frac{1}{64}\sin(8x) + C ]

So, ( \int \sin^4(2x) , dx = \frac{1}{2}x - \frac{1}{8}\sin(4x) + \frac{1}{64}\sin(8x) + 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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