# How do you integrate #int 1/sqrt(e^(2x)+12e^x-45)dx# using trigonometric substitution?

nice.. ...and the mighty Pythagorean right triangle supports this....

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Ok it work my bad

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where

Completing the square at the denominator gives

To make use of the identity

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To integrate ( \int \frac{1}{\sqrt{e^{2x} + 12e^x - 45}} , dx ) using trigonometric substitution, we can let ( e^x = \sec(\theta) ). Then, ( dx = \sec(\theta) \tan(\theta) , d\theta ). Substitute these into the integral and simplify it in terms of ( \theta ). After simplification, you should have an integral in the form ( \int \frac{\sec(\theta) \tan(\theta)}{\sqrt{\sec^2(\theta) + 12\sec(\theta) - 45}} , d\theta ). Then, use the trigonometric identity ( \sec^2(\theta) - 1 = \tan^2(\theta) ) to simplify the expression under the square root. After that, you can proceed with the integration, which typically involves completing the square and then applying standard trigonometric integration techniques. Once you integrate in terms of ( \theta ), you will need to back-substitute ( e^x = \sec(\theta) ) 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.

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