# How do you integrate #tan^4x sec^4x dx#?

When integrating a function that is a product of tangents and secants, a good strategy is to either to have

or to have

If we can do this, we can simply integrate by substitution. This may be confusing, but looking at this concrete example will help.

Undoing substitution:

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To integrate ( \tan^4(x) \sec^4(x) , dx ), you can use the substitution method. Let ( u = \tan(x) ). Then, ( du = \sec^2(x) , dx ).

Substitute ( u = \tan(x) ) and ( du = \sec^2(x) , dx ) into the integral:

[ \int \tan^4(x) \sec^4(x) , dx = \int u^4 , du ]

Integrate ( u^4 ) with respect to ( u ):

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

Now, substitute back ( u = \tan(x) ):

[ \frac{\tan^5(x)}{5} + C ]

So, the integral of ( \tan^4(x) \sec^4(x) , dx ) is ( \frac{\tan^5(x)}{5} + 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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