# How do you integrate #int e^(2x)/sqrt(-e^(2x) -81)dx# using trigonometric substitution?

See the Explanation below.

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To integrate ( \int \frac{e^{2x}}{\sqrt{-e^{2x} - 81}} , dx ) using trigonometric substitution, you can let ( e^x = 9\sec\theta ). Then proceed with the following steps:

- Find ( dx ) in terms of ( d\theta ).
- Express the integral in terms of ( \theta ).
- Use trigonometric identities to simplify the integral.
- Integrate the simplified expression with respect to ( \theta ).
- Finally, convert the result back to the variable ( x ).

This substitution helps simplify the integral by transforming it into a form that can be more easily integrated using trigonometric techniques.

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