# How do you integrate # csc^3x#?

We have:

First, rewrite the integral as follows in order to apply integration by parts:

Utilizing integration by components

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To integrate ( \csc^3(x) ), you can use trigonometric substitution. Let ( u = \csc(x) ) and ( du = -\csc(x) \cot(x) , dx ). Then, ( \int \csc^3(x) , dx ) becomes ( \int -u^2 , du ), which can be easily integrated to get ( -\frac{1}{2}u^3 + C ). Finally, substitute ( u = \csc(x) ) back in to get the final result: ( -\frac{1}{2}\csc^3(x) + C ).

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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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