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Find the integral. \int \frac{e^x}{e^x + e}dx

Question:

Find the integral.

{eq}\int \frac{e^x}{e^x + e}dx {/eq}

Indefinite Integral:

Those indefinite integrals that are formed with the substitution of the part of the integral expression in terms of the second variables will give the final result in the original variable only. But for that, we have to apply the back substitution method.

Answer and Explanation:


The indefinite integral here is:

{eq}\int \frac{e^x}{e^x + e}dx\\ {/eq}

Take the substitution of the denominator as follows:

{eq}u=e^x+e\\ \Rightarrow \:du=e^xdx\\ {/eq}

So this will have the indefinite integral of u as follows;

{eq}\int \frac{e^x}{e^x + e}dx\\=\int \frac{1}{u}du\\ =\ln \left|u\right|+c\\ =\ln \left|e^x+e\right|+C~~~~~~~~~~~~~~~~~~~~~~~\left [ \because u=e^x+e \right ] {/eq}


Learn more about this topic:

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Indefinite Integral: Definition, Rules & Examples

from Calculus: Tutoring Solution

Chapter 7 / Lesson 14
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