Integral of 1/(a^2+x^2)^(3/2) (substitution)

✅ 𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐢𝐯𝐞 𝐭𝐨 𝐜𝐡𝐞𝐜𝐤 𝐭𝐡𝐞 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧

🔧 𝐔𝐬𝐞𝐟𝐮𝐥 𝐯𝐢𝐝𝐞𝐨s

🚶 𝐒𝐭𝐞𝐩𝐬
00:00 Substitution: x=a*tan(u)
00:15 Differentiate in both sides
00:23 Substitute x and dx
00:54 Take out a^2 as common factor
01:25 Rewrite expression
01:57 Simplify expression
03:15 Integrate cos(u)
03:23 Undo substitution: u in terms of x
03:54 Use sin(arctan(x)) formula
04:10 Rewrite expression
05:01 Simplify expression
05:08 Add integration constant +C
05:14 Final answer!

🔍 𝐀𝐫𝐞 𝐲𝐨𝐮 𝐥𝐨𝐨𝐤𝐢𝐧𝐠 𝐟𝐨𝐫 𝐚 𝐩𝐚𝐫𝐭𝐢𝐜𝐮𝐥𝐚𝐫 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥? 𝐅𝐢𝐧𝐝 𝐢𝐭 𝐰𝐢𝐭𝐡 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐬𝐞𝐚𝐫𝐜𝐡𝐞𝐫:

🎓 𝐇𝐚𝐯𝐞 𝐲𝐨𝐮 𝐣𝐮𝐬𝐭 𝐥𝐞𝐚𝐫𝐧𝐞𝐝 𝐚𝐧 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝? 𝐅𝐢𝐧𝐝 𝐞𝐚𝐬𝐲, 𝐦𝐞𝐝𝐢𝐮𝐦 𝐚𝐧𝐝 𝐡𝐢𝐠𝐡 𝐥𝐞𝐯𝐞𝐥 𝐞𝐱𝐚𝐦𝐩𝐥𝐞𝐬 𝐡𝐞𝐫𝐞:

👋 𝐅𝐨𝐥𝐥𝐨𝐰 @𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬𝐟𝐨𝐫𝐲𝐨𝐮 𝐟𝐨𝐫 𝐚 𝐝𝐚𝐢𝐥𝐲 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥! 😉

𝐅𝐨𝐥𝐥𝐨𝐰 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬 𝐅𝐨𝐫𝐘𝐨𝐮

𝐃𝐨𝐧𝐚𝐭𝐞

#integralsforyou #integrals #integrationbysubstitution

[IntegralsForYou]

💪 Thumbs up if you were getting stuck in an electrostatics problem 😉👍

✅ 𝐃𝐞𝐫𝐢𝐯𝐚𝐭𝐞 𝐭𝐨 𝐜𝐡𝐞𝐜𝐤 𝐭𝐡𝐞 𝐬𝐨𝐥𝐮𝐭𝐢𝐨𝐧


🔍 𝐀𝐫𝐞 𝐲𝐨𝐮 𝐥𝐨𝐨𝐤𝐢𝐧𝐠 𝐟𝐨𝐫 𝐚 𝐩𝐚𝐫𝐭𝐢𝐜𝐮𝐥𝐚𝐫 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥? 𝐅𝐢𝐧𝐝 𝐢𝐭 𝐰𝐢𝐭𝐡 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐬𝐞𝐚𝐫𝐜𝐡𝐞𝐫:


🎓 𝐇𝐚𝐯𝐞 𝐲𝐨𝐮 𝐣𝐮𝐬𝐭 𝐥𝐞𝐚𝐫𝐧𝐞𝐝 𝐚𝐧 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝? 𝐅𝐢𝐧𝐝 𝐞𝐚𝐬𝐲, 𝐦𝐞𝐝𝐢𝐮𝐦 𝐚𝐧𝐝 𝐡𝐢𝐠𝐡 𝐥𝐞𝐯𝐞𝐥 𝐞𝐱𝐚𝐦𝐩𝐥𝐞𝐬 𝐡𝐞𝐫𝐞:


👋 𝐅𝐨𝐥𝐥𝐨𝐰 @𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬𝐟𝐨𝐫𝐲𝐨𝐮 𝐟𝐨𝐫 𝐚 𝐝𝐚𝐢𝐥𝐲 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥! 😉

[mmukulkhedekar]

anyone who came up here while getting stuck at solving electrostatics problems lol

[陳彥霖-pn]

WOW! Thanks for doing me a great favor in electromagnetics. Perfect integral performance, from Taiwan.

[benwinstanleymusic]

I too was stuck on an electrostatics problem xD

[НикитаНазаров-шэ]

Thanks for solving this integral. It helps me in electrodynamics and magnetism a lot. Best wishes from Korolyov, Russia

[jacobthrilby]

I'm in an Electromagnetism course now after taking Calc 4 almost 6 years ago. This video gives me faith that I'll be able to make it. Thank you!

[mariaclarascarpellipereira]

For those who got a little bit confused in the beginning (maybe don't remember),
derivative of a.tg(u) is a/cos²(u)du because the derivative of the tangent is the same of the derivative of (sen/cos).
This get us with the quocient rule (sen'.cos -sen.cos')/cos². This is (cos²+sen²)/cos² that is equal to 1/(cos²).

[sarathy]

Thank you so much buddy❤️...You saved my day from being stucked up in this integral while doing Electrostatics.

[juverianaeem]

I am grateful to god cuz there are people who reduce our stress considerably and they do not know what they are doing Grateful viewer from India

[garygs]

Man, you're awesome.
You're channel is wonderful
The way you solve the integrals is amazing.

[W.M.-]

Could've saved yourself the last three steps by observing that tan(arctan(x/a)) = x/a. Then you get a triangle with opposite side x, adjacent side a and hypotenuse sqrt(x^2 + a^2). The sine of this triangle is opp/hyp = x/sqrt(x^2 + a^2).

[adityavenayak]

Thanks for this integral. Helped me in electrostatics. Best wishes from India.

[artikuniyal]

Thankyou so much.. Now I can solve electrodynamics and magnetism problems easily.. Best wishes from Indian❤

[pedromoose]

Thank You! I wish I could develop the intuition to make that first substitution.

[m.espinosa]

I love the little vertical equal sign lol

[DH-sxwd]

It's a very useful video!!! Thanks for it :) (from.KOREA)

[abhirupiitkgp]

Thank you so much. Helped me a lot in electrostatics. Definitely subscribing. Love from India 😊😊

[rafaelrosante]

Thank you my friend! This type of video Helps a lot

[kushmehar]

thanks it helped me in electrostat and magnetism 🥰

[dianet]

Thanks so much for posting this! I was wondering how did you get the step you just finished at 4:08?