Solving a Problematic Integral: Tips and Tricks | Help with Difficult Integrals

[transgalactic]

i added a file with with this integral

in which i tried to solve it

how do i solve this case?

please help

 

[Schrodinger's Dog]

If the top is a derivative of the bottom in this case:-

$$\frac{e^{2x}-1}{e^{2x}+3}$$

derivative of $$\frac{d}{dx}e^{2x}-1=2e^{2x}$$

using the log rule for integrals.

$$h(x)=\frac{derivative of bottom}{bottom}=ln(bottom)$$

Since it is a derivative by a factor of

$$(e^{2x}+3)^{2/3}=ln(e^{2x}+3)^{2/3}$$

I think that's what they've done for d/dt.

$$\frac {2(e^{2x})}{e^{2x}+3}$$

 

[transgalactic]

if what you say is true than the
question must be constructed otherwise
because the derivative of the bottom is NOT what is on TOP??

i saparated them into 2 integrals the second part is the
problematic
because when
i take the whole
1/(e^2x+3) and i and take this as a complex function LAN (as shown in the file)
ln(e^2x+3)/(2*e^2x) it doesn't come out as in the book

what is the law of transforming into lan fuction exept of the case 1/x

i meen in the case of complex function
when i have 1/(something else then simple lenear stuff)?/

 

[Schrodinger's Dog]

if what you say is true than the
question must be constructed otherwise
because the derivative of the bottom is NOT what is on TOP??

i saparated them into 2 integrals the second part is the
problematic
because when
i take the whole
1/(e^2x+3) and i and take this as a complex function LAN (as shown in the file)
ln(e^2x+3)/(2*e^2x) it doesn't come out as in the book

what is the law of transforming into lan fuction exept of the case 1/x

i meen in the case of complex function
when i have 1/(something else then simple lenear stuff)?/

I think your right looking at the answer in d/dx it looks like this

$$\frac {2}{3}ln(3+e^{2x})-\frac{1}{6}ln(e^{2x})$$

A bit different.

and for $$\frac{d}{dt}$$ the paremetric, I get nothing like your books answer either? Sorry I couldn't be of more help, I don't know what they've done then? I'm sure one of the resident experts can figure it out?

$$\frac{e^{2x}+1}{e^{2x}-3}=\frac{d}{dx} of bottom=2e^{2x}$$

which is

$$\frac {1}{2}(e^{2x})$$

or $$\frac{1}{2}(top)$$

was what I was thinking, but this is apparently not right.

 

[transgalactic]

how i deside if its this is lan function

i can take every time all the function on the buttom
put it in a lan
an divide it in its derivative?

even if i got a fuction
1/(x+2)^2

its integral solves [(x+2)^-1]/-1

or i can open this equetion (a+b)^2=x^2+2*x*2+2^2
then put it all in a lan ln(x^2+2*x*2+2^2) and i will divide it by it derivative

2x+4

ln(x^2+2*x*2+2^2)/2x+4

what i need to do?
can some one make reason in this problem

 

[Mathgician]

try solving with the inverse tangent function.

1/[3((e^t/srt3)^2+1)] , try integrating that and tell me if you get what you are looking for

 

[arildno]

It is perhaps simplest to use the variable change $u=e^{x}\to{du}=\frac{du}{u}$

Then we get:
$$\int\frac{e^{2x}-1}{e^{2x}+3}dx=\int\frac{u^{2}-1}{(u^{2}+3)u}du$$
We then use partial fractions decomposition:
$$\frac{u^{2}-1}{(u^{2}+3)u}=\frac{Au+B}{u^{2}+3}+\frac{C}{u}\to{C}=-\frac{1}{3}, A=\frac{4}{3},B=0$$
Then your problem is readily solved

 

[transgalactic]

thank u very much

 

[D H]

arildno Your solution is incorect.

Arildno did fine. He did not solve the problem in full; he left the final steps up to the original poster.

The only problem with Arildno's post is the use of the word "simplest":

It is perhaps simplest to use the variable change $u=e^{x}\to{du}=\frac{du}{u}$

It is even simpler to use the variable change
$$u=e^{2x}\to{dx}=\frac1 {2u}du}$$

Then
$$\int\frac{e^{2x}-1}{e^{2x}+3}dx\to \int\frac1 2\; \frac{u-1}{u(u+3)}du$$
Decomposing,
$$\frac1 2\frac{u-1}{u(u+3)} = \frac1 6\left(\frac4{u+3}-\frac1u\right)$$
which leads to the easily integrable function
$$\int\frac1 6\left(\frac4{u+3}-\frac1u\right)du$$

 

[Mathgician]

Arildno did fine. He did not solve the problem in full; he left the final steps up to the original poster.

The only problem with Arildno's post is the use of the word "simplest":


It is even simpler to use the variable change
$$u=e^{2x}\to{dx}=\frac1 {2u}du}$$

Then
$$\int\frac{e^{2x}-1}{e^{2x}+3}dx\to \int\frac1 2\; \frac{u-1}{u(u+3)}du$$
Decomposing,
$$\frac1 2\frac{u-1}{u(u+3)} = \frac1 6\left(\frac4{u+3}-\frac1u\right)$$
which leads to the easily integrable function
$$\int\frac1 6\left(\frac4{u+3}-\frac1u\right)du$$

wrong, on the demominator, you cannot factor out u from 3, a constant. look carefully...

 

[cristo]

wrong, on the demominator, you cannot factor out u from 3, a constant. look carefully...

Which line do you think is wrong?

 

[D H]

wrong, on the demominator, you cannot factor out u from 3, a constant. look carefully...

What are you talking about? The technique is called "partial fraction decomposition". A reference: http://mathworld.wolfram.com/PartialFractionDecomposition.html"

It is easy to verify that
$$\frac1 2\frac{u-1}{u(u+3)} = \frac1 6\left(\frac4{u+3}-\frac1u\right)$$
The least common denominator of the sum on the left hand side is $u(u+3)$. Expanding the left hand side,
$$\frac1 6\left(\frac4{u+3}-\frac1u\right) = \frac1 6\left(\frac4{u+3}-\frac1u\right)\frac{u(u+3)}{u(u+3)} = \frac1 6\;\frac{4u-(u+3)}{u(u+3)} = \frac1 6\;\frac{3u-3}{u(u+3)} = \frac1 2\frac{u-1}{u(u+3)}$$
which is of course the right-hand side.

 

[Mathgician]

What are you talking about? The technique is called "partial fraction decomposition". A reference: http://mathworld.wolfram.com/PartialFractionDecomposition.html"

It is easy to verify that
$$\frac1 2\frac{u-1}{u(u+3)} = \frac1 6\left(\frac4{u+3}-\frac1u\right)$$
The least common denominator of the sum on the left hand side is $u(u+3)$. Expanding the left hand side,
$$\frac1 6\left(\frac4{u+3}-\frac1u\right) = \frac1 6\left(\frac4{u+3}-\frac1u\right)\frac{u(u+3)}{u(u+3)} = \frac1 6\;\frac{4u-(u+3)}{u(u+3)} = \frac1 6\;\frac{3u-3}{u(u+3)} = \frac1 2\frac{u-1}{u(u+3)}$$
which is of course the right-hand side.

what I am trying to say is that the substitution does not equate to the original equation if you observe the denominator, it is not hard to spot, very obvious blunder...

 

[D H]

what I am trying to say is that the substitution does not equate to the original equation if you observe the denominator, it is not hard to spot, very obvious blunder...

Point out this "obvious blunder". The only "obvious blunder" is that you apparently do not know how to use substitution to make integration easier.

Since the OP is long gone, I will finish the integration in post #10.
$$\int\frac1 6\left(\frac4{u+3}-\frac1u\right)du = \frac 1 6\left(4\log(u+3)-\log(u)\right) + C$$
This result is in terms of $u$. The desired result is in terms of the original variable $x$. Applying the substitution $u=e^{2x}$,
$$\int\frac{e^{2x}-1}{e^{2x}+3}dx = \frac 1 6\left(4\log(e^{2x}+3)-2x\right) + C = \frac 1 3\left(2\log(e^{2x}+3)-x\right) + C$$

To verify, differentiate the right-hand side wrt $x$:
$$\frac d{dx} \left(\frac 1 3\left(2\log(e^{2x}+3)-x\right)+C\right) = \frac 1 3\left(2\frac{2e^{2x}}{e^{2x}+3} - 1\right)$$
Simplifying,
$$\frac 1 3\left(2\frac{2e^{2x}}{e^{2x}+3} - 1\right) = \frac 1 3\;\frac{4e^{2x}-(e^{2x}+3)}{e^{2x}+3} = \frac{e^{2x}-1}{e^{2x}+3}$$
which of course is the function to be integrated.

 

[arildno]

It is perhaps simplest to use the variable change $u=e^{x}\to{dx}=\frac{du}{u}$

Then we get:
$$\int\frac{e^{2x}-1}{e^{2x}+3}dx=\int\frac{u^{2}-1}{(u^{2}+3)u}du$$
We then use partial fractions decomposition:
$$\frac{u^{2}-1}{(u^{2}+3)u}=\frac{Au+B}{u^{2}+3}+\frac{C}{u}\to{C}=-\frac{1}{3}, A=\frac{4}{3},B=0$$
Then your problem is readily solved

We have, by the values for A, B, C:
$$\frac{\frac{4}{3}u+0}{u^{2}+3}+\frac{\frac{-1}{3}}{u}=\frac{(\frac{4}{3}u+0)u+(-\frac{1}{3})(u^{2}+3)}{(u^{2}+3)u}=\frac{(\frac{4}{3}-\frac{1}{3})u^{2}+0u+(-\frac{1}{3})*3}{(u^{2}+3)u}=\frac{u^{2}-1}{(u^{2}+3)u}$$
as we should have.

In the future, mathgician, please remain silent when you haven't got the competence to make an informed contribution

And, DH:
I DID qualify "simplest" by the word "perhaps"..

 

[dextercioby]

Here's my treat

$$\int \frac{e^{2x}-1}{e^{2x}+3}{}dx =\int \frac{e^{2x}+3-4}{e^{2x}+3}{}dx =x-4\int \frac{e^{-2x}}{1+3e^{-2x}}{}{dx}=x+\frac{2}{3}\int\frac{d\left(1+3e^{-2x}\right)}{1+3e^{-2x}}$$

$$=x+\ln\left(\left(1+3e^{-2x}\right)^{\frac{2}{3}}\right) +C$$

Simpler or simplest ?

 

[arildno]

Straightforward, but not necessarily simplest.