How do you integrate #int 1/sqrt(e^(2x) +100)dx# using trigonometric substitution?
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To integrate ( \int \frac{1}{\sqrt{e^{2x} + 100}} , dx ) using trigonometric substitution:

Recognize that ( e^{2x} + 100 ) can be rewritten as ( 100  e^{2x} ).

Let ( e^x = 10\sec(\theta) ). Then, ( dx = 10\sec(\theta)\tan(\theta) , d\theta ).

Substitute ( e^x ) and ( dx ) with the expressions obtained in step 2 into the integral.

Rewrite ( e^{2x} + 100 ) in terms of ( \theta ) using the trigonometric identity ( \sec^2(\theta)  1 = \tan^2(\theta) ).

Simplify the expression to integrate in terms of ( \theta ).

Integrate the simplified expression with respect to ( \theta ).

Substitute back the original variable ( x ) using the trigonometric identity ( e^x = 10\sec(\theta) ).

Simplify the expression obtained after substituting back to get the final result.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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