Integration of Logarithmic Functions: int [ e^x (e^x + 1)^2 ] dx

[Eigendorf]

Hello Again!!

I've just finished my first semester of calculus.. YAY!

I still have a few questions on some topics from the class that I didn't fully understand so I'll probably post a few over the christmas break.

One question I had was why a result was different when I used a different integration approach.

The question asked:

Find the Indefinite Integral of

f(x) = \(\displaystyle \displaystyle \int e^x \cdot \left( e^x + 1 \right)^2 \: dx \)


This is fairly straightforward using the chain rule.

I set

\(\displaystyle \displaystyle u = e^x + 1 \)

Then:

\(\displaystyle \displaystyle du = e^x dx \)


Substituting back into the integral:

\(\displaystyle \displaystyle \int u^2 \: du \)


Applying the Integral:

F(x) = \(\displaystyle \displaystyle \dfrac{u^3}{3} + C\)


Finally replacing with the X term.

The answer is:

F(x) = \(\displaystyle \displaystyle \dfrac{(e^x + 1)^3}{3} + C\)

I've verified this answer on my graphing calculator.


My question is:

Why does the answer come out different when I expand the function first, then integrate?

Ie..

f(x) = \(\displaystyle \displaystyle \int e^x \cdot \left( e^x + 1 \right)^2 \: dx \) = \(\displaystyle \displaystyle \int e^{3x} + 2e^{2x} + e^x \: dx \)

So.. Integrating the right expression should equal the same as integrating the left expression.


f(x) = \(\displaystyle \displaystyle \int e^{3x} + 2e^{2x} + e^x \: dx \)

And applying the integral

F(x) = \(\displaystyle \displaystyle \dfrac{e^{3x}}{3} + e^{2x} + e^x + C\)


I got two different results here but I can see that if I set \(\displaystyle c = \dfrac{1}{3}\) it makes this answer the same as the first integral, assuming c = 0 for the first integral.

I don't understand why there is a difference in these integration constants and what strategy I should use to choose between which method to integrate with.

Ie.. is there a way to tell which is more appropriate: Integrating with the chain rule or expanding first then integrating.

Thanks again for all of the help.


P.S. I'm still trying to figure this LaTex stuff out. Is there a way to adjust the font size of the math functions displayed with LaTex?

 

[Deleted member 4993]

Hello Again!!

I've just finished my first semester of calculus.. YAY!

I still have a few questions on some topics from the class that I didn't fully understand so I'll probably post a few over the christmas break.

One question I had was why a result was different when I used a different integration approach.

The question asked:

Find the Indefinite Integral of

f(x) = \(\displaystyle \displaystyle \int e^x \cdot \left( e^x + 1 \right)^2 \: dx \)


This is fairly straightforward using the chain rule.

I set

\(\displaystyle \displaystyle u = e^x \)

Then:

\(\displaystyle \displaystyle du = e^x dx \)


Substituting back into the integral:

\(\displaystyle \displaystyle \int u^2 \: du \) ..................Incorrect............ it should be \(\displaystyle \displaystyle \int (u+1)^2 \: du \)


Applying the Integral:

F(x) = \(\displaystyle \displaystyle \dfrac{u^3}{3} + C\)


Finally replacing with the X term.

The answer is:

F(x) = \(\displaystyle \displaystyle \dfrac{(e^x + 1)^3}{3} + C\)

I've verified this answer on my graphing calculator.


My question is:

Why does the answer come out different when I expand the function first, then integrate?

Ie..

f(x) = \(\displaystyle \displaystyle \int e^x \cdot \left( e^x + 1 \right)^2 \: dx \) = \(\displaystyle \displaystyle \int e^{3x} + 2e^{2x} + e^x \: dx \)

So.. Integrating the right expression should equal the same as integrating the left expression.


f(x) = \(\displaystyle \displaystyle \int e^{3x} + 2e^{2x} + e^x \: dx \)

And applying the integral

F(x) = \(\displaystyle \displaystyle \dfrac{e^{3x}}{3} + e^{2x} + e^x + C\)


I got two different results here but I can see that if I set \(\displaystyle c = \dfrac{1}{3}\) it makes this answer the same as the first integral, assuming c = 0 for the first integral.

I don't understand why there is a difference in these integration constants and what strategy I should use to choose between which method to integrate with.

Ie.. is there a way to tell which is more appropriate: Integrating with the chain rule or expanding first then integrating.

Thanks again for all of the help.


P.S. I'm still trying to figure this LaTex stuff out. Is there a way to adjust the font size of the math functions displayed with LaTex?

You should designate two different constants (e.g. C₁ and C₂) for two different methods of attack. As long as those two integrated functions have same derivatives - both will be acceptable answer.

 

[tkhunny]

f(x) = \(\displaystyle \displaystyle \int e^x \cdot \left( e^x + 1 \right)^2 \: dx \)

\(\displaystyle \displaystyle u = e^x \)

Try \(\displaystyle u = e^{x} + 1\).

Please read and heed my signature. This has other implications, such as, if you get two different answers, one of them most likely is incorrect. Emphasis on "unique".

 

[Eigendorf]

Try \(\displaystyle u = e^{x} + 1\).

Please read and heed my signature. This has other implications, such as, if you get two different answers, one of them most likely is incorrect. Emphasis on "unique".

You are correct.

I actually made a typo when I created this post. I was concentrating so hard on this LaTex stuff that I didn't double check my notes were correct.

I had performed the integration I used \(\displaystyle \displaystyle u = e^x + 1 \)

I updated my first post.

Everything in my first post was operating under the assumption that \(\displaystyle \displaystyle u = e^x + 1 \)


I think the different answers is just a matter of perspective. The integration constant can be combined with any other constant because the indefinite integral is really just a family of possible functions.

So, I could make the C2 = to 1/3 for the second integral and the C1 = 0 for the first integral and the point where the two different integrals meet is the correct answer. Although this his just my assumption, I'm hoping you guys can give me a better perspective on this.

edit.. (editing as I'm thinking about this)

Or Looking at this another way..

I can pull 1/3 out of F(x) and combine it with the integration constant and then the integral of 2 functions are equal.

Like.. if I pull 1/3 out of this

F(x) = \(\displaystyle \displaystyle \dfrac{(e^x + 1)^3}{3} + C\)

and combine it with the integration constant..

F(x) then becomes

F(x) = \(\displaystyle \displaystyle \dfrac{e^{3x}}{3} + e^{2x} + e^x + C\)

So, hmm, maybe I could just go with the strategy of combining all constants with the integration constant when I'm comparing different integration techniques.

 

[Dr.Peterson]

I think the different answers is just a matter of perspective. The integration constant can be combined with any other constant because the indefinite integral is really just a family of possible functions.

So, I could make the C2 = to 1/3 for the second integral and the C1 = 0 for the first integral and the point where the two different integrals meet is the correct answer. Although this his just my assumption, I'm hoping you guys can give me a better perspective on this.

This is the best way to see it, I think (though I'm not sure what you mean by "meet"); the fact that a solution is a family of functions, not just one function, is the essence of what Khan said. The constant is arbitrary; constants you get by different methods need not be the same, and the two will be equal for some pairing of the constants. In this case, C2 is just C1 + 1/3.

That is, what look like two different functions (implying an error) are really the same function. This happens quite often in integration, and is in fact the reason teachers insist on your always writing the "+ C".