# How do you integrate #int tan^5(2x)sec^2(2x)#?

We'll have to make a trigonometric substitution here, and we have tangents and secants, so we're looking for either a tangent with an odd power or a secant with an event power.

So:

Plugging that back into the original integral:

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To integrate ( \int \tan^5(2x)\sec^2(2x) , dx ), you can use trigonometric identities and integration by substitution. Start by rewriting ( \tan^5(2x) ) as ( (\sec^2(2x) - 1)^2 \tan(2x) ). Then, let ( u = \sec(2x) ) and ( du = 2\sec(2x) \tan(2x) , dx ). After substituting, the integral becomes ( \frac{1}{2} \int (u^2 - 1)^2 , du ), which can be integrated term by term. Finally, substitute back ( \sec(2x) ) for ( u ) to obtain the final result.

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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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