# How do you integrate #int xsqrt(x-1)# by parts?

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Please see below.

Rewrite algebraically to taste. I like the answer above, but others might prefer

Or

Or some equivalent expression.

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

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

Let's denote:

[ u = \sqrt{x-1} ] [ dv = x , dx ]

Differentiate ( u ) to get ( du ) and integrate ( dv ) to get ( v ):

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

Now we substitute these into the integration by parts formula:

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

Simplify and integrate the second term:

[ = \frac{1}{2}x^2\sqrt{x-1} - \frac{1}{4} \int \frac{x^2}{\sqrt{x-1}} , dx ]

To integrate ( \frac{x^2}{\sqrt{x-1}} ), we can use the substitution method. Let ( t = \sqrt{x-1} ), then ( x = t^2 + 1 ) and ( dx = 2t , dt ). Substitute these into the integral:

[ = \frac{1}{2}x^2\sqrt{x-1} - \frac{1}{4} \int \frac{(t^2+1)^2}{t} \cdot 2t , dt ] [ = \frac{1}{2}x^2\sqrt{x-1} - \frac{1}{2} \int (t^4 + 2t^2 + 1) , dt ] [ = \frac{1}{2}x^2\sqrt{x-1} - \frac{1}{2} \left( \frac{1}{5}t^5 + \frac{2}{3}t^3 + t \right) + C ]

Finally, substitute back ( t = \sqrt{x-1} ) to get the final answer:

[ \int x\sqrt{x-1} , dx = \frac{1}{2}x^2\sqrt{x-1} - \frac{1}{2} \left( \frac{1}{5}(\sqrt{x-1})^5 + \frac{2}{3}(\sqrt{x-1})^3 + \sqrt{x-1} \right) + 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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