# What is the net area between #f(x) = x^2-xln(x^2-1) # and the x-axis over #x in [2, 4 ]#?

To find the area we must integrate the function then apply the limits to the integral:

Integrating the first term is straightforward but I suspect people may have trouble integrating the second term so let's look in more detail how we do that:

We can then apply the substitution to obtain:

So now using integration by parts gives us:

So we have effectively shown that:

We can now proceed with the original question stated, that is:

Using our integral that we just we found out:

Now putting the limits in:

Simplifying:

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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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