# How do you integrate #int sec^2x/(4-tan^2x)^(3/2)# by trigonometric substitution?

We have:

Thus, the tangent is opposite over adjacent, or:

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To integrate the given expression using trigonometric substitution, we can let ( \tan{x} = u ), so ( \sec^2{x} ) becomes ( 1 + u^2 ). Then, we substitute ( \sec^2{x} = 1 + u^2 ) and ( \tan^2{x} = u^2 ) into the expression. After substitution, we end up with ( \int \frac{1}{(4 - u^2)^{3/2}} , du ). Next, we can use a trigonometric identity to simplify this integral. By letting ( u = 2\sin{\theta} ), we can rewrite the expression in terms of ( \theta ). After making this substitution, we can integrate with respect to ( \theta ) and then convert back to the variable ( x ).

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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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