I'd like some clarification on integrating f(ax+b)

Astronomer_X

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Hi! Today in class we were learning how to integrate f(ax+b). Most of the exercise has been fine, and I understand all the standard function rules from differentiation. However:

My text book outlined this method of integrating a function in the form of (ax+b)n (Look at Question B, the second one explained):


So far, I understood the premise well; it's similar to the basic integration I learnt last year, but we do not divide by (n+1) after raising the power. Fine. I then tried to apply it to this question:


As you can see, I followed the same logic to be applied. However, this is what the solution/answer actually is:


I can see that they expanded the equation first, but should the same method as outlined in the example not work as well? If so, why/why not? Because imagine if the question I failed at was raised to the power of 3, or 4 or 5- expanding that first would be ridiculously time consuming. Did I just not apply the chain rule right?
 

MarkFL

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I would say the reason you didn't get the correct answer is because the derivative of the "inner" function is not simply a constant. You were trying to apply a method for:

\(\displaystyle f(x)=(ax+b)^n\)

to a function of the form:

\(\displaystyle g(x)=\left(ce^x+d\right)^n\)

This causes an issue with having the correct differential in your integral when you attempt to substitute.
 

Astronomer_X

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I would say the reason you didn't get the correct answer is because the derivative of the "inner" function is not simply a constant. You were trying to apply a method for:

\(\displaystyle f(x)=(ax+b)^n\)

to a function of the form:

\(\displaystyle g(x)=\left(ce^x+d\right)^n\)

This causes an issue with having the correct differential in your integral when you attempt to substitute.
So could I ask how I would apply it to g(x) in your example? Or is that not possible for that type of function?
 

MarkFL

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So could I ask how I would apply it to g(x) in your example? Or is that not possible for that type of function?
The method you've been shown would not work for \(g(x)\). It would work for this function:

\(\displaystyle h(x)=ae^{bx}\left(ce^{bx}+d\right)^n\)

Here, we could let:

\(\displaystyle y=\left(ce^{bx}+d\right)^{n+1}\implies dy=(n+1)bce^{bx}\left(ce^{bx}+d\right)^n\,dx\)

Do you see now that the differential \(dy\) is just some constant times \(h(x)\,dx\)?
 

Astronomer_X

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I see. So, I should just go about a g(x) type question by expanding first?
 

MarkFL

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I see. So, I should just go about a g(x) type question by expanding first?
Yes, and hopefully you won't be given a problem of that type where \(n\) is large, although in theory, the binomial theorem could be used for the expansion. :)
 
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