How did Griffith check Stoke's theorem in this case?

[garylau]

<Moderator's note: Moved from a technical forum, so homework template missing>

Sorry
i have one question to ask
how to check the v.dl part in this problem
i cannot do this problem as it is too hard to integrate the equation

How did griffith get this long-horrible equation(see the orange circle)?
it sounds unreasonable and too hard to get
and is it possible that there are
other faster ways to check the v.dl part in this problem?

thank you

 

[Dragon27]

You can just take both integrals (one from each term in the numerator) from the table of integrals. Here's a list of integrals of irrational functions on the wiki. Or you can spend some time trying to derive those integrals yourself using the standard methods (but you don't need to do it each and every time - that's what the tables are for), as a useful exercise. Try to change the variable to some convenient trigonometric function to get rid of those square roots for starters.

 

[garylau]

You can just take both integrals (one from each term in the numerator) from the table of integrals. Here's a list of integrals of irrational functions on the wiki. Or you can spend some time trying to derive those integrals yourself using the standard methods (but you don't need to do it each and every time - that's what the tables are for), as a useful exercise. Try to change variables to some convenient trigonometric function to get rid of those square roots for starters.

thank

How did he prove this formula?

 

[Dragon27]

thank
How did he prove this formula?

What kind of integration techniques are you familiar with?

 

[garylau]

What kind of integration techniques are you familiar with?

the more simple one
but
i havn't derived this formula(the two circles that i emphasize) before and not familiar with this formula
it looks quiet tedious

do you know what technique the Wiki is using to claim these statements

thank

 

[Dragon27]

it looks quiet tedious

It isn't tedious at all, if you use the proper trigonometric substitution (you can even do it in your head after some training, though the paper is still less error-prone).
How about some khan academy tutorials?
https://www.khanacademy.org/math/integral-calculus/integration-techniques#trig-substitution

 

[garylau]

It isn't tedious at all, if you use the proper trigonometric substitution (you can even do it in your head after some training, though the paper is still less error-prone).
How about some khan academy tutorials?
https://www.khanacademy.org/math/integral-calculus/integration-techniques#trig-substitution

It isn't tedious at all, if you use the proper trigonometric substitution (you can even do it in your head after some training, though the paper is still less error-prone).
How about some khan academy tutorials?
https://www.khanacademy.org/math/integral-calculus/integration-techniques#trig-substitution

thank you
however
Khan didnt derive this formula which included x^2?
where is these Two terms coming from

 

[Dragon27]

Well, if you went through the tutorial, you should be able to derive this by yourself (the substitution is the same).
How about you start doing this integral and show us where you get stuck?