What Techniques Can Simplify Integrating \(\frac{e^x}{e^{2x} + 1}\)?

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In summary, an integral is a mathematical concept used to find the area under a curve in a graph. To solve an integral, techniques such as substitution, integration by parts, or partial fractions can be used. The "e" in "e^x" represents Euler's number, a mathematical constant used in many equations. There is a difference between definite and indefinite integrals, with definite integrals having specific limits and indefinite integrals resulting in an equation with a constant term. To solve the integral e^x/(e^2x+1), the substitution method can be used by letting u = e^x and solving with the inverse tangent function. The final answer will be e^x * arctan(e^x) +
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tmt1
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I have this integral

$$\int_{}^{}\frac{e^x}{{e}^{2x} + 1} \,dx$$

And I'm not sure how to approach this. I've tried u-substitution a few ways, but it seems to go to a dead end. I'm not sure how to apply partial fractions, trig-substitution, or integration by parts to this problem.
 
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I would look at:

\(\displaystyle u=e^x\,\therefore\,du=e^x\,dx\)

And so now you have:

\(\displaystyle \int \frac{1}{u^2+1}\,du\)
 

FAQ: What Techniques Can Simplify Integrating \(\frac{e^x}{e^{2x} + 1}\)?

What is an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is the inverse operation of a derivative and is used to find the original function when given its derivative.

How do you solve an integral?

To solve an integral, you can use techniques such as substitution, integration by parts, or partial fractions. It is also helpful to have a good understanding of basic integration rules and properties.

What does the "e" in "e^x" represent?

The "e" in "e^x" represents the mathematical constant known as Euler's number. It is approximately equal to 2.71828 and is commonly used in many mathematical and scientific equations.

What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration, while an indefinite integral does not. This means that a definite integral will give a numerical value, while an indefinite integral will give an equation with a constant term that needs to be solved for.

How do you solve the integral e^x/(e^2x+1)?

To solve this integral, you can use the substitution method by letting u = e^x. This will change the integral to 1/(u^2+1), which can then be solved using the inverse tangent function. The final answer will be e^x * arctan(e^x) + C, where C is the constant of integration.

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