# How do you find the antiderivative of #int tan^3xsecx dx#?

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To find the antiderivative of (\int \tan^3(x) \sec(x) , dx), we can use integration by parts. Let (u = \tan(x)) and (dv = \sec(x) , dx). Then, (du = \sec^2(x) , dx) and (v = \ln|\sec(x) + \tan(x)|).

Now, applying the integration by parts formula: [ \int u , dv = uv - \int v , du ]

We get: [ \int \tan^3(x) \sec(x) , dx = \tan(x) \ln|\sec(x) + \tan(x)| - \int \ln|\sec(x) + \tan(x)| \sec^2(x) , dx ]

The remaining integral is non-elementary, so it cannot be expressed in terms of elementary functions. Thus, the antiderivative of (\int \tan^3(x) \sec(x) , dx) involves an expression containing both (\tan(x)) and (\ln|\sec(x) + \tan(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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