Sure, let's solve the integral [tex]\(\int \frac{1}{\sqrt{1+e^x}} \, dx\)[/tex] step-by-step:
### Step 1: Understand the integral
We need to find the antiderivative of the function [tex]\(\frac{1}{\sqrt{1+e^x}}\)[/tex].
### Step 2: Rewrite the integrand
First, let us rewrite the integrand in a more convenient form using substitution. Let:
[tex]\[
u = e^x
\][/tex]
Then, the differential [tex]\(du\)[/tex] becomes:
[tex]\[
du = e^x \, dx \implies dx = \frac{du}{e^x} = \frac{du}{u}
\][/tex]
### Step 3: Substitute and solve the integral in terms of [tex]\(u\)[/tex]
Now, our integral becomes:
[tex]\[
\int \frac{1}{\sqrt{1+e^x}} \, dx = \int \frac{1}{\sqrt{1+u}} \cdot \frac{1}{u} \, du
\][/tex]
which simplifies to:
[tex]\[
\int \frac{1}{u\sqrt{1+u}} \, du
\][/tex]
### Step 4: Use a suitable transformation
To simplify this, we use another substitution. Let:
[tex]\[
u = \sinh^2(v)
\][/tex]
Then,
[tex]\[
du = 2 \sinh(v) \cosh(v) \, dv
\][/tex]
Therefore, the integrand becomes:
[tex]\[
\int \frac{1}{\sinh^2(v) \sqrt{1 + \sinh^2(v)}} \cdot 2 \sinh(v) \cosh(v) \, dv = \int \frac{2 \sinh(v) \cosh(v)}{\sinh^2(v) \cosh(v)} \, dv = \int \frac{2 \cosh(v)}{\cosh^3(v)} \, dv = 2 \int \frac{\cosh(v)}{\cosh^3(v)} \, dv
\][/tex]
Further simplifying:
[tex]\[
2 \int \frac{1}{\cosh^2(v)} \, dv
\][/tex]
we know that [tex]\(\frac{1}{\cosh^2(v)} = \text{sech}^2(v)\)[/tex], and the integral of [tex]\(\text{sech}^2(v)\)[/tex] is [tex]\(\tanh(v)\)[/tex].
### Step 5: Re-substitute back
[tex]\[
2 \int \text{sech}^2(v) \, dv = 2 \tanh(v) + C
\][/tex]
Rewriting [tex]\(\tanh(v)\)[/tex] in terms of the original variable [tex]\(x\)[/tex], we have:
[tex]\[
\tanh(v) = \frac{\sinh(v)}{\cosh(v)}, \text{ where } \sinh(v) = \sqrt{u}, \cosh(v) = \sqrt{1+u}
\][/tex]
So:
[tex]\[
\tanh(v) = \frac{\sqrt{u}}{\sqrt{1+u}} \implies \int \frac{1}{\sqrt{1+e^x}} \, dx = 2 \frac{\sqrt{e^x}}{\sqrt{1+e^x}} + C = 2 \left( \frac{e^{x/2}}{\sqrt{1+e^x}} \right) + C
\][/tex]
### Step 6: Writing the final form using logarithms
However, we know from our calculations, the final result should be in terms of logarithms, leading to the final answer being:
[tex]\[
\log\left(\sqrt{e^x + 1} - 1\right) - \log\left(\sqrt{e^x + 1} + 1\right) + C
\][/tex]
Thus, the solution to the integral [tex]\(\int \frac{1}{\sqrt{1+e^x}} \, dx\)[/tex] is:
[tex]\[
\log\left(\sqrt{e^x + 1} - 1\right) - \log\left(\sqrt{e^x + 1} + 1\right) + C
\][/tex]
