# How do you find the antiderivative of #int (csc^3x) dx#?

Thus:

Short proof of the integral of cosecant:

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To find the antiderivative of (\int \csc^3(x) , dx), you can use trigonometric identities and integration techniques. One method is to rewrite (\csc^3(x)) as (\csc(x) \cdot (\csc^2(x))). Then, perform a substitution letting (u = \csc(x)). This leads to the integral becoming (\int \frac{1}{u^3} , du), which can be integrated using the power rule for integration. The antiderivative is (-\frac{1}{2u^2} + C), where (C) is the constant of integration. Finally, substitute back (u = \csc(x)) to get the final answer: (-\frac{1}{2\csc^2(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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