# How do you find the antiderivative of #dx/(e^(2x)-9)^(1/2)#?

We have to make this substitution:

so:

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To find the antiderivative of ( \frac{dx}{(e^{2x}-9)^{1/2}} ), you can use a substitution method. Let ( u = e^{2x} - 9 ). Then, ( du = 2e^{2x}dx ), which can be simplified to ( \frac{du}{2} = e^{2x}dx ).

Now, substitute ( u = e^{2x} - 9 ) and ( \frac{du}{2} = e^{2x}dx ) into the integral:

( \int \frac{dx}{(e^{2x}-9)^{1/2}} = \frac{1}{2} \int \frac{du}{u^{1/2}} )

This integral is now in a form that can be directly integrated using the power rule for integration:

( \frac{1}{2} \int u^{-1/2} du = \frac{1}{2} \cdot \frac{u^{1/2}}{1/2} + C )

Finally, substitute back for ( u ) to get the final antiderivative:

( \frac{1}{2} \cdot 2(e^{2x}-9)^{1/2} + C = (e^{2x}-9)^{1/2} + 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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