"Evaluate this integral: ∫ex1+exdx" was answered by students like you

High School
Calculus and Analysis
Calculus 2
Applications of integrals

rheisf

Answered question

2022-01-03

Evaluate this integral:

\int \frac{e^{x}}{1 + e^{x}} d x

Answer & Explanation

eskalopit

Beginner2022-01-04Added 31 answers

Step 1
Given,
The integral

\int \frac{e^{x}}{1 + e^{x}} d x

We have to evaluate the integral.
Step 2
Lets

Virginia Palmer

Beginner2022-01-05Added 27 answers

\int \frac{e^{x}}{1 + e^{x}} d x

We put the expression

\exp (x)

under the differential sign, i.e.:

e^{x} d x = d (e^{x}), t = e^{x}

Then the original integral can be written as follows:

\int \frac{1}{t + 1} d t

\int \frac{1}{x + 1} d x

Calculate the table integral:

\int \frac{1}{x + 1} d x = \ln (x + 1)

Answer:

\ln (x + 1) + C

To write the final answer, it remains to substitute

\exp (x)

instead of t.

\ln (e^{x} + 1) + C

Vasquez

Expert2022-01-07Added 669 answers

$\begin{matrix} G i v e n : \\ \int \frac{e^{x}}{e^{x} + 1} d x \\ S u b s t i t u t i o n u = e^{x} + 1 \\ = \int \frac{1}{u} d u \\ = \ln (u) \\ = \ln (e^{x} + 1) \\ A n s w e r : \\ = \ln (e^{x} + 1) + C \end{matrix}$

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