# How do you integrate #int e^x/(7+e^(2x))# by trigonometric substitution?

write it in a format that we can look to integrate more easily

The integral in terms of u is now

differentiating using the quotient rule

The integral is now of the form

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To integrate ( \frac{e^x}{7+e^{2x}} ) by trigonometric substitution, first let ( u = e^x ). Then, ( du = e^x , dx ). Rewrite the integral in terms of ( u ) to get ( \int \frac{1}{7+u^2} , du ). This is a standard form for trigonometric substitution. Let ( u = \sqrt{7} \tan{\theta} ). Then, ( du = \sqrt{7} \sec^2{\theta} , d\theta ). Substitute ( u ) and ( du ) into the integral and simplify. This results in ( \frac{1}{\sqrt{7}} \int \frac{\sec^2{\theta}}{7+7\tan^2{\theta}} , d\theta ). Now, use the trigonometric identity ( \sec^2{\theta} = 1 + \tan^2{\theta} ) to simplify the expression further. After simplification, the integral becomes ( \frac{1}{\sqrt{7}} \int \frac{1}{14} , d\theta ). Integrate to get ( \frac{\theta}{\sqrt{7}} + C ). Substitute ( \theta = \arctan{\frac{u}{\sqrt{7}}} ) back into the equation. Thus, the final answer is ( \frac{\arctan{\frac{e^x}{\sqrt{7}}}}{\sqrt{7}} + 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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