Integration by part relating to cosine and sine

D

dBanji

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Hi,
I have tried integrating this function using integration by part but seem to behave like an endless cycle of sin(t)cos(t) that needs another integration by part. the function is the integration of t^6cos^2(t^7)sin(t^7).
I made u=t^7 and got the integration of cos^2(u)sin(u)du.
I used integration udv=uv - integration vdu but ended up with a cycle of the integration of the product of sine cosine needing another udv method.
I then tried substituting cos^2(u) with 1-sin^2(u) which worked well until I got to the point 1/7 multiplied by the integration of sin(u) - sin^3(u). I could not go past this stage. Can somebody help, please?
I really did not want to seek help but couldn't get past this stage. The reason being I do not want to come here seeking only help especially because I was asked to show if I really did attempt to solve a problem yesterday and referred back to the rules of the forum. I did everything I could before coming here.

Also, sorry, I do not have an app to type this more accurately (mathematically). In addition, my phone does not connect with my computer so if I take a shot of the handwritten work, I have to send it to email, then download, and then attach it before sending. It is a lot of processes. I hope you can understand the questions.

Many thanks.
 
Last edited:
dBanji said:
Hi,
I have tried integrating this function using integration by part but seem to behave like an endless cycle of `sin(t)cos(t)` that needs another integration by part. the function is the integration of `t^6cos^2(t^7)sin(t^7)`.
I made `u=t^7` and got the integration of `cos^2(u)sin(u)du`.
I used `int udv=uv - int vdu` but ended up with a cycle of the integration of the product of sine cosine needing another `udv` method.
I then tried substituting `cos^2(u)` with `1-sin^2(u)` which worked well until I got to the point `1/7` multiplied by the integration of `sin(u) - sin^3(u)`. I could not go past this stage. Can somebody help, please?
I really did not want to seek help but couldn't get past this stage. The reason being I do not want to come here seeking only help especially because I was asked to show if I really did attempt to solve a problem yesterday and referred back to the rules of the forum. I did everything I could before coming here.

Also, sorry, I do not have an app to type this more accurately (mathematically). In addition, my phone does not connect with my computer so if I take a shot of the handwritten work, I have to send it to email, then download, and then attach it before sending. It is a lot of processes. I hope you can understand the questions.

Many thanks.
Click to expand...

First, your way of writing the expressions is perfectly readable; it's one of several ways we recommend. In fact, if you put backquotes around each of your expressions, you'll find that the site formats it nicely for you with no additional changes other than replacing the full word "integration" with "int". I did that for you above.

Second, repeating integration by parts can work; there are several typical scenarios that you learn if you study this long enough.

But third, have you tried a substitution, v = cos(u)?

Integration is an art, and it is not uncommon to have to try several approaches before you find something that works. It's also not uncommon to need a little nudge in the right direction.
 
Thank you for your time and help. I have tried v=cos(u) still getting the same thing. I also tried another method attached herewith.
I have been able to log in using my phone so I’m able to take a shot directly.
Thank you for helping to edit my previous posting.
I have been trying this problem all night.
 

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HINT: What is the derivative of \(\cos^3(t^7)~?\)
 
First, why do you write dx, when the variable is t? That may get you in trouble?

Second, in the first image, how does `cos^2(t^7)` become `cos^2(u)`, when `u=sin(t^7)`? You are being careless about the substitution!

Third, you should be using `u=cos(t^7)`.

Similar mistakes are made in the second image, though I didn't look closely at it because parts are not needed.
 
pka said:
HINT: What is the derivative of \(\cos^3(t^7)~?\)
Click to expand...
It's derivative is
Dr.Peterson said:
First, why do you write dx, when the variable is t? That may get you in trouble?

Second, in the first image, how does `cos^2(t^7)` become `cos^2(u)`, when `u=sin(t^7)`? You are being careless about the substitution!

Third, you should be using `u=cos(t^7)`.

Similar mistakes are made in the second image, though I didn't look closely at it because parts are not needed.
Click to expand...
ok. let me try again.
thanks
 
dBanji said:
It's derivative is

ok. let me try again.
thanks
Click to expand...
Thank you very much. I think I can see where I’ve gone wrong. I chose the wrong u-substitution and that got me into trouble. Sorry for the confusion about dt and dx I guess I’m so used to dx that I forget to change that. I’ll take note of that. I have here what I think is the right answer. Would you please check for me?
Thank you once again.

I forgot to add Plus C i.e. +C at the end. Just a note.

Thanks.
 

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Last edited:
I’m sorry, I think I created a new thread after sending that. The question looks odd but there are other similar questions like that like three. I just do not have a clue how to proceed
 
Here is the graph generated using Desmond
 

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