# How do you evaluate the integral #int x^2arctanx#?

The answer is

Therefore,

so,

Finally, we have

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We will use the following Rule of Integration by Parts (IBP) :

The Later Integral of J has been directly obtained using the

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To evaluate the integral ( \int x^2 \arctan(x) ), you can use integration by parts. Integration by parts formula is ( \int u , dv = uv - \int v , du ).

Let ( u = \arctan(x) ) and ( dv = x^2 , dx ).

Then, ( du = \frac{1}{1+x^2} , dx ) and ( v = \frac{x^3}{3} ).

Now, apply the integration by parts formula:

( \int x^2 \arctan(x) , dx = \frac{x^3}{3} \arctan(x) - \int \frac{x^3}{3} \cdot \frac{1}{1+x^2} , dx )

Now, to evaluate the remaining integral, you might need to use a substitution or another method depending on the complexity of the expression.

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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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