Solving integral with substitution

[Shannabel]

Homework Statement

find the integral from 0 to ln2 of (e^x-1)/(e^x+1)

Homework Equations

The Attempt at a Solution

I really didn't know what to do, but the book had a hint suggesting i multiply the numerator and denominator by e^-x and then use an appropriate substitution..
[((e^x)-1)*(e^-x)]/[((e^x)+1)*(e^-x)]
= (1-(e^-x))/(1+(e^-x))
= (1-(e^x))/(1+(e^x))
but now i can't think of an appropriate substitution. help?

 

[c.dube]

How did you get from:
$\frac{1-e^{-x}}{1+e^{-x}}$
To:
$\frac{1-e^{x}}{1+e^{x}}$

 

[Shannabel]

How did you get from:
$\frac{1-e^{-x}}{1+e^{-x}}$
To:
$\frac{1-e^{x}}{1+e^{x}}$

apparently i just made up my own rules of calculus.. you're right, i can't do that.
so I'm just at
[1-e^(-x)]/[1+e^(-x)]

 

[c.dube]

Hahah I don't know how your book wants you to do it but try splitting it into two integrals.

 

[Shannabel]

Hahah I don't know how your book wants you to do it but try splitting it into two integrals.

how can i split that?

 

[c.dube]

Into two fractions - you have addition on top - and then attack them both separately.

 

[HallsofIvy]

The first thing I would do is divide:
$$\frac{e^x- 1}{e^x+ 1}= 1- \frac{2}{e^x+ 1}$$

Then, for the second integral, let $u= e^x+ 1$

 

[Shannabel]

The first thing I would do is divide:
$$\frac{e^x- 1}{e^x+ 1}= 1- \frac{2}{e^x+ 1}$$

Then, for the second integral, let $u= e^x+ 1$

lol... what did you divide by to get that?
and also, if i use u=e^x, then i get du=e^xdx, and i don't have anything to cancel the extra e^x with..?

 

[obafgkmrns]

I'm not sure what multiplying top & bottom by e^-x does for you, but if you multiply top & bottom by e^(-x/2), you'll probably recognize it as the hyperbolic tangent of x/2.

 

[Bohrok]

lol... what did you divide by to get that?
and also, if i use u=e^x, then i get du=e^xdx, and i don't have anything to cancel the extra e^x with..?

You can use long division to divide them, similar to normal division of numbers. Or you can write it as $$\frac{e^x - 1}{e^x + 1} = \frac{e^x + 1 - 2}{e^x + 1} = \frac{e^x + 1}{e^x + 1} - \frac{2}{e^x + 1} = 1 - \frac{2}{e^x + 1}$$
If you want to follow Hall's suggestion, you should multiply the fraction by e^-x/e^-x and then use a different substitution.

I would follow c.dube's suggestion by splitting the fraction in two $$\frac{e^x - 1}{e^x + 1} = \frac{e^x}{e^x + 1} - \frac{1}{e^x + 1}$$The first one has an easy substitution, but you'll want to multiply the second by e^-x/e^-x before using a substitution.

 

[Shannabel]

You can use long division to divide them, similar to normal division of numbers. Or you can write it as $$\frac{e^x - 1}{e^x + 1} = \frac{e^x + 1 - 2}{e^x + 1} = \frac{e^x + 1}{e^x + 1} - \frac{2}{e^x + 1} = 1 - \frac{2}{e^x + 1}$$
If you want to follow Hall's suggestion, you should multiply the fraction by e^-x/e^-x and then use a different substitution.

I would follow c.dube's suggestion by splitting the fraction in two $$\frac{e^x - 1}{e^x + 1} = \frac{e^x}{e^x + 1} - \frac{1}{e^x + 1}$$The first one has an easy substitution, but you'll want to multiply the second by e^-x/e^-x before using a substitution.

ookay so i ended up with
ln(e^(x)+1)+ln(e^(-x)+1)... is that right?

 

[Shannabel]

You can use long division to divide them, similar to normal division of numbers. Or you can write it as $$\frac{e^x - 1}{e^x + 1} = \frac{e^x + 1 - 2}{e^x + 1} = \frac{e^x + 1}{e^x + 1} - \frac{2}{e^x + 1} = 1 - \frac{2}{e^x + 1}$$
If you want to follow Hall's suggestion, you should multiply the fraction by e^-x/e^-x and then use a different substitution.

I would follow c.dube's suggestion by splitting the fraction in two $$\frac{e^x - 1}{e^x + 1} = \frac{e^x}{e^x + 1} - \frac{1}{e^x + 1}$$The first one has an easy substitution, but you'll want to multiply the second by e^-x/e^-x before using a substitution.

ookay so i ended up with
ln(e^(x)+1)+ln(e^(-x)+1) between 0 and ln2
= ln((e^x+1)/(e^-x)+1)) between 0 and ln2
= ln ((e^ln2+1)/(e^-ln2+1))-ln((e^0+1)/e^0+1))
= ln(3/-1)-ln(1)
= ln(-3)

.. which is very wrong, obviously.
where did i go wrong?

 

[Bohrok]

The two logs are being added, so you need to multiply their arguments and not divide them.

 

[Shannabel]

The two logs are being added, so you need to multiply their arguments and not divide them.

ahhh! of course!
so now i get

ln((e^x+1)*(e^-x+1))
= [ln((e^ln2+1)*(e^-ln2+1))]-[ln(e^0+1)*(e^0+1)]
= ln(-3)-ln(4)
= ln(-3/4)

closer? but still not right...?

 

[Bohrok]

e^-ln2 ≠ -2
Make sure you use the rules of logs and exponents correctly when you simplify.

 

[c.dube]

Look at the part $e^{-\ln2}$ and how you could re-express that. Then run with it.
EDIT: Haha as Bohrok said.

 

[Shannabel]

e^-ln2 ≠ -2
Make sure you use the rules of logs and exponents correctly when you simplify.

e^-ln2= (1/2)??

 

[Bohrok]

Yes!

 

[Shannabel]

yes!

got it! :):)

 

[obafgkmrns]

You guys are making this much too difficult! Note that the integrand is tanh(x/2) which is in turn sinh(x/2)/cosh(x/2). Substitute u = sinh(x/2) and recall that the derivative of sinh is cosh. The integrand then becomes 1/u, which integrates to ln(u). Resolve the substitution and voila.

 

[Shannabel]

You guys are making this much too difficult! Note that the integrand is tanh(x/2) which is in turn sinh(x/2)/cosh(x/2). Substitute u = sinh(x/2) and recall that the derivative of sinh is cosh. The integrand then becomes 1/u, which integrates to ln(u). Resolve the substitution and voila.

my prof left the part of the course with function like sinh and cosh out of the course this year.