# How do you integrate #int xsec^-1x# by integration by parts method?

We now have, by the integration by parts formula,

Hopefully this helps!

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To integrate ( \int x\sec^{-1}(x) ) using integration by parts, we use the formula:

[ \int u , dv = uv - \int v , du ]

Let ( u = \sec^{-1}(x) ) and ( dv = x , dx ). Then, we have:

[ du = \frac{1}{|x|\sqrt{x^2 - 1}} , dx ] [ v = \frac{x^2}{2} ]

Now, applying the integration by parts formula:

[ \int x \sec^{-1}(x) , dx = \frac{x^2}{2}\sec^{-1}(x) - \int \frac{x^2}{2} \cdot \frac{1}{|x|\sqrt{x^2 - 1}} , dx ]

This integral is not straightforward, but it can be solved by further manipulations or using different methods like trigonometric substitution.

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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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