Evaluate the integral? : # int \ x^2/(sqrt(x^2-25))^5 \ dx#
Substitute:
to have:
use now the trigonometric identity:
To undo the substitution note that:
Then:
and undoing the substitution:
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# int \ x^2/(sqrt(x^2-25))^5 \ dx = -1/75 \ x^3/( sqrt(x^2 -25) )^3 + C #
Compare the denominator to the trig identity:
In an attempt to reduce the denominator to something simper.
So we want to find:
Thus:
Which we can integrate by observation as:
And so we conclude that:
If we refer back to our earlier substitution, we note:
Using this to reverse the earlier substitution we get:
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To evaluate the integral ( \int \frac{x^2}{(\sqrt{x^2-25})^5} , dx ), we can use the substitution method. Let ( u = \sqrt{x^2 - 25} ). Then, ( x^2 = u^2 + 25 ) and ( dx = \frac{u}{\sqrt{u^2 + 25}} , du ).
Substituting these into the integral:
[ \int \frac{x^2}{(\sqrt{x^2-25})^5} , dx = \int \frac{u^2 + 25}{u^5} \left(\frac{u}{\sqrt{u^2 + 25}} \right) , du ]
[ = \int \frac{u^3 + 25u}{u^5 \sqrt{u^2 + 25}} , du ]
[ = \int \frac{u^2 + 25}{u^5 \sqrt{u^2 + 25}} , du + \int \frac{25u}{u^5 \sqrt{u^2 + 25}} , du ]
[ = \int \frac{1}{u^3 \sqrt{u^2 + 25}} , du + 25 \int \frac{1}{u^4 \sqrt{u^2 + 25}} , du ]
Now, these integrals can be evaluated using standard integral formulas. After finding the antiderivatives, you'll need to back-substitute ( u = \sqrt{x^2 - 25} ) to express the final result in terms of ( x ).
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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.
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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