jamessinatraaa

2021-12-27

Find the indefinite integral.

$\int \frac{{e}^{-x}}{1+{e}^{-x}}dx$

lalilulelo2k3eq

Beginner2021-12-28Added 38 answers

Step 1

Integration is summation of discrete data. The integral is calculated for the functions to find their area, displacement, volume, that occurs due to combination of small data.

Integration is of two types definite integral and indefinite integral. Indefinite integral are defined where upper and lower limits are not given, whereas in definite integral both upper and lower limit are there.

Step 2

The given integrand is$\int \frac{{e}^{-x}}{1+{e}^{-x}}dx$ . Consider the denominator of the integrand $1+{e}^{-x}$ , let it be equal to u. Differentiate the denominator of integrand with respect to x .

$u=1+{e}^{-x}$

$\frac{du}{dx}=\frac{d(1={e}^{-x})}{dx}$

$=\frac{d1}{dx}+\frac{d{e}^{-x}}{dx}$

$=0+{e}^{-x}(-1)$

$du=-1\left({e}^{-x}\right)dx$

$(-1)du={e}^{-x}dx$ ...(1)

Step 3

Substitute value of$u=1+{e}^{-x}$ and ${e}^{-x}dx=(-1)du$ from equation (1) in $\int \frac{{e}^{-x}}{1+{e}^{-x}}dx$

$\int \frac{{e}^{-x}}{1+{e}^{-x}}dx=\int \frac{(-1)du}{u}$

$-1\mathrm{ln}\left|u\right|+C$

$=-1\mathrm{ln}|1+{e}^{-x}|+C$ (As $1+{e}^{-x}=u$ )

Therefore, integration of$\int \frac{{e}^{-x}}{1+{e}^{-x}}dx$ is $-1\mathrm{ln}|1={e}^{-x}|+C$

Integration is summation of discrete data. The integral is calculated for the functions to find their area, displacement, volume, that occurs due to combination of small data.

Integration is of two types definite integral and indefinite integral. Indefinite integral are defined where upper and lower limits are not given, whereas in definite integral both upper and lower limit are there.

Step 2

The given integrand is

Step 3

Substitute value of

Therefore, integration of

Mary Nicholson

Beginner2021-12-29Added 38 answers

We put the expression exp (x) under the sign of the differential, i.e.:

Then the original integral can be written as follows: We put the

Expression

Then the original integral can be written as follows:

Calculate the tabular integral:

Answer:

Since previously we made a change of variables, then instead of u we substitute 1 / t.

To write down the final answer, it remains to substitute exp (x) instead of t.

karton

Expert2022-01-04Added 613 answers

Answer:

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