Domain of antiderivative problem

Jan 2017
23
0
Europe
So I have the following problem:

\(\displaystyle \int \frac{4e^x}{(1-e^{2x})^2} dx\)

I realize this can be computed a simple U-substitution (followed by rationalizing the function), however I wanted to get in some practice with trig substitutions.

\(\displaystyle \int \frac{4e^x}{(1-e^{2x})^3} dx =\int \frac{4e^x}{(1-(e^x)^2)^3} dx \)

Now I set the following:

\(\displaystyle e^x = sin(t) \rightarrow x = ln(sin(t)) \rightarrow dx = \frac{1}{sin(t)} * cos(t) dt\)

Substituting:

\(\displaystyle \int \frac{4e^x}{(1-e^{2x})^2} dx = 4\int \frac{sin(t)}{(1-sin(t)^2)^3} * \frac{1}{sin(t)} * cos *dt = 4\int \frac{1}{cos(t)^5}\)
\(\displaystyle 4\int sec(t)^5 dt = 4\int sec(t)^3 sec(t)^2 = sec(t)^3 tan(t) - \int 3 sec(t)^2 tan(t) dt\)

So we get:

\(\displaystyle \int \frac{4e^x}{(1-e^{2x})^2} dx = 4(sec(t)^3 tan - \frac{3}{2}tan(t)^2 + C\)

To express it in terms of X:

\(\displaystyle cos(t) = \sqrt{1-(e^x)^2}\rightarrow sec(t) = \frac{1}{\sqrt{1-(e^x)^2}} \rightarrow tan(t) = \frac{e^x}{\sqrt{1-(e^x)^2}} \)

Finally substituting back, the final answer becomes:

\(\displaystyle 4((\frac{1}{\sqrt{1-(e^x)^2}})^3 \frac{e^x}{\sqrt{1-(e^x)^2}} - \frac{3}{2}(\frac{e^x}{\sqrt{1-(e^x)^2}})^2) + C\)

My problem is that the domain of this function is only x < 0 (for real numbers). What should I make of that?
 
Last edited:

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
Note the domain of the original integrand is all reals, $x \ne 0$

By doing the trig sub, $e^x = \sin{t}$, you've restricted the domain to $-\infty < x < 0$ from the very start ...
 
Jan 2017
23
0
Europe
So the only way this trig substitution would work here is if this were a definite integral problem with a range that falls somewhere between \(\displaystyle -\infty\) and 0 ?
 

Plato

MHF Helper
Aug 2006
22,490
8,653
So I have the following problem:
\(\displaystyle \int \frac{4e^x}{(1-e^{2x})^2} dx\)
I realize this can be computed a simple U-substitution (followed by rationalizing the function), however I wanted to get in some practice with trig substitutions.
My problem is that the domain of this function is only x < 0 (for real numbers). What should I make of that?
If instead use $u=e^x$ and get $\dfrac{du}{(1-u^2)^3}$
Now use partial fractions, SEE HERE, to eliminate the domain issues.
 
Jan 2017
23
0
Europe
Thank you for the comment Plato. I'm aware of that, but the problem I'm having is that I'm not fully understanding how the domain changes with the various substitutions, so I'm trying to work the problem in ways that will expose me to domain-related issues. This is a general weakness of mine when it comes to integrals and I'm trying to improve it.
 
Last edited:

Plato

MHF Helper
Aug 2006
22,490
8,653
I'm not fully understanding how the domain changes with the various substitutions, so I'm trying to work the problem in ways that will expose me to domain-related issues.
That is the whole point of using $u=e^x$. That substitution does not change the domain.
Still $x\ne 0$ makes $(1-u)\ne 0$.

Do you understand how making a u-substitution in a definite integral changes the limits of integration?
 
Jan 2017
23
0
Europe
Do you understand how making a u-substitution in a definite integral changes the limits of integration?
I believe I do. However up until now I was always substituting back to the original integration term, so I wasn't actually calculating the new limits when making the substitution; so far it had been working out for me and this excercise over here is the first time I actually noticed an issue with the domain of the function can arise when substituting (it's not something that I recall was mentioned in any of the online materials I've been learning from).

So essentially whenever I'm about to make a substitution of any kind, I should first make sure that the substitution I am planning to use does change the range of the function? And if it does change the range, then I need to use a different method of integration?
 
Jan 2017
23
0
Europe
Hello,

Just bumping this in case anyone could answer the question in my last post. Would appreciate the help as I'm still not sure about this.
 
Jan 2017
23
0
Europe
What exactly is your question?
Hi Plato, this is the question I was referring to:

So essentially whenever I'm about to make a substitution of any kind, I should first make sure that the substitution I am planning to use does change the range of the function? And if it does change the range, then I need to use a different method of integration?