Error in solving $\int \sqrt{1 + e^x} dx$ .

Question

I want to solve this integral for $1 + e^x \ge 0$

$$\int \sqrt{1 + e^x} dx$$

I start by parts $$\int \sqrt{1 + e^x} dx = x\sqrt{1 + e^x} - \int x \frac{e^x}{2\sqrt{1 + e^x}} dx $$

Substitute $\sqrt{1 + e^x} = t \implies dt = \frac{e^x}{2\sqrt{1 + e^x}} dx$

So I remain with $$x\sqrt{1 + e^x} - \int \ln((t-1)(t+1)) dt = x\sqrt{1 + e^x} + 2t +(-t+1)\ln(t-1)+ (-t+1)\ln(t+1) $$

that is

$$(x+2)\sqrt{1 + e^x} + (-\sqrt{1 + e^x} + 1) \ln(+\sqrt{1 + e^x} - 1) + (-\sqrt{1 + e^x} + 1) \ln(+\sqrt{1 + e^x} + 1) + C $$

But the solution should be $$2\sqrt{1 + e^x} + \ln(+\sqrt{1 + e^x} - 1) - \ln(+\sqrt{1 + e^x} + 1) $$

Where is the mistake hidden?

Answer 1

Where is the mistake hidden?

In the following part :

So I remain with $$x\sqrt{1 + e^x} - \int \ln((t-1)(t+1)) dt = x\sqrt{1 + e^x} + 2t +(-t+1)\ln(t-1)+ (-t+1)\ln(t+1) $$

It should be the following (a sign mistake) : $$x\sqrt{1 + e^x} - \int \ln((t-1)(t+1)) dt = x\sqrt{1 + e^x} + 2t +(-t+1)\ln(t-1)+ (-t\color{red}{-}1)\ln(t+1)\color{red}{+C} $$

that is $$x\sqrt{1 + e^x} + 2\sqrt{1 + e^x} +(-\sqrt{1 + e^x}+1)\ln(\sqrt{1 + e^x}-1)$$$$+ (-\sqrt{1 + e^x}-1)\ln(\sqrt{1 + e^x}+1)+C$$ $$=(x+2)\sqrt{1+e^x}-\sqrt{1+e^x}\ (\ln (\sqrt{1+e^x}-1)+\ln(\sqrt{1+e^x}+1))+\ln(\sqrt{1+e^x}-1)$$$$-\ln(\sqrt{1+e^x}+1)+C$$ $$=2\sqrt{1+e^x}+\ln(\sqrt{1+e^x}-1)-\ln(\sqrt{1+e^x}+1)+C$$

Answer 2

Let $x = \ln t$. Then $dx = dt/t$ and we have $$\int \sqrt{1 + e^x} \, dx = \int \frac{\sqrt{1 + t}}t \, dt.$$

Now let $u = t + 1$. Then $du = dt$ and we have $$\int \frac{\sqrt{1+t}}t \, dt = \int\frac{\sqrt{u}}{u-1} \, du.$$

Now let $v = \sqrt{u}$. Then $u = v^2$ and so $du = 2v \, dv$, and we have $$\int\frac{\sqrt{u}}{u-1} \, du = 2\int\frac{v^2}{v^2 - 1} \, dv.$$

Consider then: \begin{align*} 2\int\frac{v^2}{v^2 - 1} \, dv &= 2\int\frac{v^2 \color{red}{-1+1}}{v^2 - 1} \, dv \\[0.3cm] &= 2\int\frac{v^2-1}{v^2-1} \, dv + 2\int\frac{dv}{v^2-1}\\[0.3cm] &= 2v + \ln(v-1) - \ln(v+1) + C \end{align*}

Back substitute: $$ 2v + \ln(v-1) - \ln(v+1) + C = 2\sqrt{u} + \ln(\sqrt{u}-1) - \ln(\sqrt{u}+1) + C $$

Back substitute: $$ 2\sqrt{u} + \ln(\sqrt{u}-1) - \ln(\sqrt{u}+1) + C = 2\sqrt{t+1} + \ln(\sqrt{t+1}-1) - \ln(\sqrt{t+1}+1) + C $$

Back substitute $t = e^x$ to finally get: $$ \int \sqrt{1+e^x} \, dx = 2\sqrt{e^x+1} + \ln(\sqrt{e^x+1}-1) - \ln(\sqrt{e^x+1}+1) + C $$

Answer 3

with $$t=\sqrt{1+e^x}$$ we get $$x=\ln(t^2-1)$$ and from here $$dx=\frac{2t}{t^2-1}dt$$ and our integral will be $$\int\frac{2t^2}{t^2-1}dt$$

Answer 4

HINT:

Let $\sqrt{1+e^x}=y\implies e^x=y^2-1\implies e^x\ dx=2y\ dy$

$$\int\sqrt{1+e^x}\ dx=\int\dfrac{2y^2}{y^2-1}dy$$

Now $y^2=y^2-1+1$

$$\dfrac{2y^2}{y^2-1}=2+\dfrac{y+1-(y-1)}{y^2-1}=2+\dfrac1{y-1}-\dfrac1{y+1}$$