Irvin Dukes

Answered

2021-12-20

Evaluate the following integrals.

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

Answer & Explanation

kalupunangh

Expert

2021-12-21Added 29 answers

Step 1

We have to evaluate the integral:

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

Rewriting the integral,

$\int \frac{{e}^{x}}{{e}^{2x}+2{e}^{x}+1}dx=\int \frac{{e}^{x}}{\left({e}^{x}\right)+2{e}^{x}+1}dx$

This will be solved by substitution method,

so assuming$t={e}^{x}$

differentiating,

$\frac{dt}{dx}=\frac{d{e}^{x}}{dx}$

$\frac{dt}{dx}={e}^{x}$

$dt={e}^{x}dx$

Step 2

Substituting above values in the integral,

$\int \frac{{e}^{x}}{{\left({e}^{x}\right)}^{2}+2{e}^{x}+1}dx=\int \frac{dt}{{t}^{2}+2t+1}$

$=\int \frac{dt}{{\left(t\right)}^{2}+2\left(t\right)\left(1\right)+{\left(1\right)}^{2}}$

$=\int \frac{dt}{{(t+1)}^{2}}$

$=-\frac{1}{t+1}+C$ (since $\int \frac{1}{{(x+a)}^{2}}=-\frac{1}{x+a}+C$ )

Where, C is an arbitrary constant.

Hence, value of integral is$-\frac{1}{{e}^{x}+1}+C$ .

We have to evaluate the integral:

Rewriting the integral,

This will be solved by substitution method,

so assuming

differentiating,

Step 2

Substituting above values in the integral,

Where, C is an arbitrary constant.

Hence, value of integral is

Barbara Meeker

Expert

2021-12-22Added 38 answers

It is required to calculate:

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

Substitution$u={e}^{x}\Rightarrow \frac{du}{dx}={e}^{x}\Rightarrow dx={e}^{-x}du$ , we use:

$e}^{2x}={u}^{2$

$=\int \frac{1}{{u}^{2}+2u+1}du$

Let us factorize:

$=\int \frac{1}{{(u+1)}^{2}}du$

Substitution$v=u+1\Rightarrow \frac{dv}{du}=1\Rightarrow du=dv:$

$=\int \frac{1}{{v}^{2}}dv$

Integral of a power function:

$\int {v}^{n}dv=\frac{{v}^{n+1}}{n+1}$ at n=-2:

$=-\frac{1}{v}$

Reverse replacement v=u+1:

$=-\frac{1}{u+1}$

Reverse replacement$u={e}^{x}:$

$=-\frac{1}{{e}^{x}+1}$

Problem solved:

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

$=-\frac{1}{{e}^{x}+1}+C$

Substitution

Let us factorize:

Substitution

Integral of a power function:

Reverse replacement v=u+1:

Reverse replacement

Problem solved:

nick1337

Expert

2021-12-28Added 573 answers

Let us put the expression exp (x) under the sign of the differential, ie:

Then the original integral can be written as follows:

Let's

use the method of decomposition into simplest elements. Let us expand the function into the simplest terms:

Equate the numerators and take into account that the coefficients at the same powers of x , standing on the left and on the right, must coincide:

1=A(x+1)+B

x: A=0

1: A+B=1

Solving it, we find:

A=0; B=1;

We calculate the table integral:

NSK

Answer:

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

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