I really appreciated when you explained the thinking behimd the trigonometric substition. i have studied this for a good times and it was the first time it clicked with me. very simple and elegant touch, thanks as always Graeme!
+Sean Thrasher Hi Sean, and lovely to make contact again! I am glad that you found my explanation helpful. It always bothered me when people would 'drag some substitution (or adjustment factor) from mid-air' with no explanation as to how they came to choose that particular arrangement. It is encouraging to find others who want to know and understand, too. Thank you for letting me know that my explanation was of value to you.
Very clever way of finding the solution, that dividing numerator and denominator by cos squared of x. We were trying to come close to tan x, indirectly through sec x. That led us to substitution that produced tan as a solution. No escaping from tan in this integral, huh? Yet another multilayered workout and solution. Thank you for explaining it to us, dear Graeme.
+MrVoayer Thank you, MrVoayer. The basic rule is that, if you can see a way of creating sec²x.dx in the integral, then you should consider substituting u = tanx. An alternative way of evaluating this integral would be to use the double angle formula cos2θ = cos²θ - sin²θ = 2cos²θ - 1 to replace the cos²x with (cos2x + 1)/2. There are a few obstacles to overcome using this method, but it would be a good exercise for students to evaluate this integral in both ways. I am glad that you liked the video and thank you for your faithfulness in providing encouraging feedback.
Hello, wjrasmussen666. I certainly do intend to complete the entire set (and revisit a few that I have already covered and show alternative approaches). I am sorry that it has taken so long and that there has been such an extended hiatus, but the last few years have been extremely difficult and have required that I devote my attentions elsewhere. Hopefully, I will have my 'studio' functional again and have created sufficient time by the middle of this year and will resume posting videos then. Thank you for asking and kind regards to you.
+Sean Thrasher My reply (below) still stands, friend. No need to apologise at all. It is very encouraging to be informed that I have achieved my goal. That is, to explain WHY we look for certain features and take particular steps in solving integrals. There IS a lot of ingenuity and experience involved, but there are also some very important principles to learn so that, mostly, it is no mystery why a particular path has been followed. Thank you, again, and best wishes to you, Sean.
Hi Harish, It would be wrong because cosx is not simply a linear function of x. The equation ∫1/(1 + f(x)²).dx = arctan(f(x)) + C only applies if f(x) is linear. In fact, even then, a coefficient of x complicates matters just a bit. I am sure that you are familiar with integrals such as ∫1/(1 + 4x²).dx. As soon as you being inserting other functions, you create a 'chain rule' situation where the derivative also must appear somewhere. This means that you have an entirely new link of integral. If you want the understand this better, calculate the derivative of arctan(cosx) and compare it with what you have in the integral under discussion here. As always, it is good to see that you have an enquiring mind and are asking very good questions. You will learn a lot this way. Warm regards, Graeme
I really appreciated when you explained the thinking behimd the trigonometric substition. i have studied this for a good times and it was the first time it clicked with me. very simple and elegant touch, thanks as always Graeme!
+Sean Thrasher Hi Sean, and lovely to make contact again!
I am glad that you found my explanation helpful. It always bothered me when people would 'drag some substitution (or adjustment factor) from mid-air' with no explanation as to how they came to choose that particular arrangement.
It is encouraging to find others who want to know and understand, too.
Thank you for letting me know that my explanation was of value to you.
Very clever way of finding the solution, that dividing numerator and denominator by cos squared of x. We were trying to come close to tan x, indirectly through sec x. That led us to substitution that produced tan as a solution. No escaping from tan in this integral, huh? Yet another multilayered workout and solution. Thank you for explaining it to us, dear Graeme.
+MrVoayer Thank you, MrVoayer.
The basic rule is that, if you can see a way of creating sec²x.dx in the integral, then you should consider substituting
u = tanx.
An alternative way of evaluating this integral would be to use the double angle formula
cos2θ = cos²θ - sin²θ = 2cos²θ - 1 to replace the cos²x with (cos2x + 1)/2. There are a few obstacles to overcome using this method, but it would be a good exercise for students to evaluate this integral in both ways.
I am glad that you liked the video and thank you for your faithfulness in providing encouraging feedback.
Any chance you are going to do the rest of these?
Hello, wjrasmussen666.
I certainly do intend to complete the entire set (and revisit a few that I have already covered and show alternative approaches). I am sorry that it has taken so long and that there has been such an extended hiatus, but the last few years have been extremely difficult and have required that I devote my attentions elsewhere.
Hopefully, I will have my 'studio' functional again and have created sufficient time by the middle of this year and will resume posting videos then.
Thank you for asking and kind regards to you.
i apologise, my previous comment I realised referred to #27
+Sean Thrasher My reply (below) still stands, friend. No need to apologise at all.
It is very encouraging to be informed that I have achieved my goal. That is, to explain WHY we look for certain features and take particular steps in solving integrals. There IS a lot of ingenuity and experience involved, but there are also some very important principles to learn so that, mostly, it is no mystery why a particular path has been followed.
Thank you, again, and best wishes to you, Sean.
Sir, a doubt
for ∫1/(1 + cos²x).dx
would arctan(cosx) be wrong? if so why?
Hi Harish,
It would be wrong because cosx is not simply a linear function of x. The equation ∫1/(1 + f(x)²).dx = arctan(f(x)) + C only applies if f(x) is linear. In fact, even then, a coefficient of x complicates matters just a bit. I am sure that you are familiar with integrals such as ∫1/(1 + 4x²).dx.
As soon as you being inserting other functions, you create a 'chain rule' situation where the derivative also must appear somewhere. This means that you have an entirely new link of integral. If you want the understand this better, calculate the derivative of arctan(cosx) and compare it with what you have in the integral under discussion here.
As always, it is good to see that you have an enquiring mind and are asking very good questions. You will learn a lot this way.
Warm regards,
Graeme
Thank you very much sir :)
So the method to do : ∫1/(1 + 4x²).dx this would be to substitute x=tanθ right?
You are welcome, Harish.