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?
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?
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