# How do you find the integral of #tan^2(x) * sec^3(x) dx#?

See the explanation section, below.

We get

which brings us to

and

Thus, as of right now, we have

Thus, we will now conclude with

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To integrate ( \tan^2(x) \cdot \sec^3(x) ) with respect to ( x ), you can use integration by parts. Let ( u = \tan(x) ) and ( dv = \sec^3(x) dx ). Then, ( du = \sec^2(x) dx ) and ( v = \frac{1}{2}\tan(x)\sec(x) + \frac{1}{2}\ln|\sec(x) + \tan(x)| ).

Applying the integration by parts formula ( \int u , dv = uv - \int v , du ), we get:

[ \int \tan^2(x) \cdot \sec^3(x) , dx = \frac{1}{2}\tan(x)\sec(x) + \frac{1}{2}\ln|\sec(x) + \tan(x)| - \int \sec^2(x) \cdot \left(\frac{1}{2}\tan(x)\sec(x) + \frac{1}{2}\ln|\sec(x) + \tan(x)|\right) , dx ]

[ = \frac{1}{2}\tan(x)\sec(x) + \frac{1}{2}\ln|\sec(x) + \tan(x)| - \frac{1}{2} \int \tan(x)\sec^3(x) , dx - \frac{1}{2} \int \sec(x)\ln|\sec(x) + \tan(x)| , dx ]

The integral ( \int \tan(x)\sec^3(x) , dx ) can be evaluated using substitution, and ( \int \sec(x)\ln|\sec(x) + \tan(x)| , dx ) can be evaluated using integration by parts again.

This process may need to be repeated until you get integrals that you can directly evaluate. Once you've evaluated those integrals, you can substitute the results back into the original equation to find the final answer.

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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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