# How do you integrate #x*arctan(x) dx#?

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To integrate ( x \cdot \text{arctan}(x) , dx ), you can use integration by parts. Let ( u = \text{arctan}(x) ) and ( dv = x , dx ). Then, ( du = \frac{1}{1+x^2} , dx ) and ( v = \frac{1}{2}x^2 ).

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

[ \int x \cdot \text{arctan}(x) , dx = \frac{1}{2}x^2 \cdot \text{arctan}(x) - \frac{1}{2} \int \frac{x^2}{1+x^2} , dx ]

Now, perform the integral ( \int \frac{x^2}{1+x^2} , dx ) using a substitution method, such as ( x = \tan(\theta) ). After integrating, you can substitute ( x ) back in to get the final result.

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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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