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

✍🏼 integralsforyou.com - Integral of 1/(2+x^2) - How to integrate it step by step using the substitution method!
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00:00 Take out the 2 as common factor
00:16 Take 1/2 outside the integral and write x^2/2 as (x/sqrt(2))^2
00:36 Substitution: u=x/sqrt(2)
00:46 Differentiate in both sides
00:59 Substitute x/sqrt(2) and dx
01:20 Take sqrt(2) outside the integral
01:31 Simplify fraction and integrate 1/(1+u^2)
01:52 Undo substitution: u in terms of x
02:04 Add integration constant +C
02:13 Final answer!
02:16 See more!
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✍🏼 integralsforyou.com ✍🏼 🔍 𝐀𝐫𝐞 𝐲𝐨𝐮 𝐥𝐨𝐨𝐤𝐢𝐧𝐠 𝐟𝐨𝐫 𝐚 𝐩𝐚𝐫𝐭𝐢𝐜𝐮𝐥𝐚𝐫 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥? 𝐅𝐢𝐧𝐝 𝐢𝐭 𝐰𝐢𝐭𝐡 𝐭𝐡𝐞 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥 𝐬𝐞𝐚𝐫𝐜𝐡𝐞𝐫: ► Integral searcher 👉integralsforyou.com/integral-searcher 🎓 𝐇𝐚𝐯𝐞 𝐲𝐨𝐮 𝐣𝐮𝐬𝐭 𝐥𝐞𝐚𝐫𝐧𝐞𝐝 𝐚𝐧 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝? 𝐅𝐢𝐧𝐝 𝐞𝐚𝐬𝐲, 𝐦𝐞𝐝𝐢𝐮𝐦 𝐚𝐧𝐝 𝐡𝐢𝐠𝐡 𝐥𝐞𝐯𝐞𝐥 𝐞𝐱𝐚𝐦𝐩𝐥𝐞𝐬 𝐡𝐞𝐫𝐞: ► Integration by parts 👉integralsforyou.com/integration-methods/integration-by-parts ► Integration by substitution 👉integralsforyou.com/integration-methods/integration-by-substitution ► Integration by trig substitution 👉integralsforyou.com/integration-methods/integration-by-trig-substitution ► Integration by Weierstrass substitution 👉integralsforyou.com/integration-methods/integration-by-weierstrass-substitution ► Integration by partial fraction decomposition 👉integralsforyou.com/integration-methods/integration-by-partial-fraction-decomposition 👋 𝐅𝐨𝐥𝐥𝐨𝐰 @𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬𝐟𝐨𝐫𝐲𝐨𝐮 𝐟𝐨𝐫 𝐚 𝐝𝐚𝐢𝐥𝐲 𝐢𝐧𝐭𝐞𝐠𝐫𝐚𝐥! 😉 📸 instagram.com/integralsforyou 🙋‍♂ 𝐃𝐨𝐧𝐚𝐭𝐞: ▶ Join the Integrals ForYou community: kzbin.info/door/NLRwiQSPlAn_hiEM2yWIwgjoin

[@biridun9999]

dude you have all the answers i am looking for you are the best.

[@IntegralsForYou]

Glad to hear that! Thanks for your comment, I really appreciate when you say how useful is my channel! Enjoy it! ❤❤

[@IntegralsForYou]

By the way, I recently opened a new channel about derivatives, if you subscribe it will help me to make it grow! www.youtube.com/@DerivativesForYou Thanks!

[@ohsayitdittooo]

Can you pls make one video on basics for calculus, im so bad at it and idk from where to start TT

[@adilsonsantosdasmerces7894]

Perfect!

[@IntegralsForYou]

Thank you! 💪

[@BridgetMukwazhi-yz9ui]

Brilliant❤

[@IntegralsForYou]

Thank you! ❤

[@Gamer-rs6ow]

We can see the formula of Integration of 1 / ( a^2 + x^2 ) where a is √2 and x is x. We can directly write answer, can't we ? Without using substitution.

[@pratyushsingh1211]

exactly...why is there a need to do what he has done..lmao

[@IntegralsForYou]

@@pratyushsingh1211 Hi! When I do my videos I try to not use formulas. In this case, if you are allowed to use the integral of 1/(a^2+x^2) formula I'm ok with it, but if you are not allowed to use it or you just want more details about how to solve this integral without using the formula then you have the steps in the video 😉

[@Loots1]

i remember when you were a tiny channel, happy to see your success :)

[@IntegralsForYou]

Thank you! When you come back after these years and say this kind of comments you make my day! I love you! ❤

[@Loots1]

​@@IntegralsForYou although I dont watch your videos as much, i still see your ig posts every day and most of the time break out the ol' pen and paper and get to work for fun! i love you more

[@IntegralsForYou]

@@Loots1 Thank you thank you thank you! ❤

[@alejandropulidorodriguez9723]

splendid

[@IntegralsForYou]

Thank you! 😊

[@Gamer-rs6ow]

Can you solve Integration of 1 / ( e^x + e^-x ) with voice over or in such a way it become easy

[@IntegralsForYou]

Hi! I did the video here: kzbin.info/www/bejne/faOcg4N7eN9rnrc . It has no voice but I think you will understand every step 😉. If not, you can ask me in a comment and I'll be happy to answer 💪

[@user-gx1ir2oi3j]

Please Can you solve this integration /Cos²(sinx) dx

[@IntegralsForYou]

Hi! I'm sorry but this integral is a non-elementary integral, it cannot be expressed in terms of finite standard functions...

[@Snow_Leopard_Uncia_uncia]

Как найти int.( e^ (1/x))dx ?

[@IntegralsForYou]

Hi! The integral of e^(1/x) is a non-elementary integral, it cannot be expressed in terms of finite standard functions....

[@Snow_Leopard_Uncia_uncia]

Да , пожалуй. Yeah, certainly, maybe. Неберущийся ,так сказать.

[@Snow_Leopard_Uncia_uncia]

👋🐅🐆. Why are you by year's sign?

[@Snow_Leopard_Uncia_uncia]

I'm a Tiger ( b.1962)

[@02.alomiaalejandra37]

Como se llama la propiedad que aplica?

[@IntegralsForYou]

Hola! En este video estoy aplicando el método de sustitución o cambio de variable 😉

[@02.alomiaalejandra37]

@@IntegralsForYou¿ pero porqué x^2 cambia a (×/squirt 2)^2?

[@IntegralsForYou]

@@02.alomiaalejandra37 Porque el objetivo es usar la siguiente fórmula: Integral de f'(x)/(1+[f(x)]^2) dx = arctan(f(x)) + C (donde f'(x) es la derivada de f(x)) Concretamente, para f(x)=x y f'(x)=1 obtenemos: Integral de 1/(1+x^2) dx = arctan(x) + C Como nosotros no tenemos un 1 en el denominador, debemos manipular la expresión para conseguir ese 1+(algo)^2. En nuestro caso, tenemos un 2+(algo)^2 así que sacamos factor común del 2 para obtener el 1 y luego, dado que x^2/2 no es (algo)^2 sino (algo)^2/2, metemos el 2 del denominador dentro para que finalmente ese "algo" sea x/sqrt(2). Si prefieres usar la fórmula con f(x), la integral se resolvería así: Integral de 1/(2+x^2) dx = = Integral de 1/2(1+x^2/2) dx = = Integral de (1/2)/(1+x^2/2) dx = = Integral de (1/2)/(1+(x/sqrt(2))^2) dx = = Integral de (1/sqrt(2))*(1/sqrt(2))/(1+(x/sqrt(2))^2) dx = = (1/sqrt(2))*Integral de (1/sqrt(2))/(1+(x/sqrt(2))^2) dx = Ahora ya tenemos f(x)=x/sqrt(2) al cuadrado y su derivada f'(x)=1/sqrt(2) en el numerador: = (1/sqrt(2))*arctan(x/sqrt(2)) + C Espero haberme explicado bien y haberte ayudado 💪

[@Snow_Leopard_Uncia_uncia]

33

[@IntegralsForYou]

❤

[@bhoopalsinghRoheria]

Kya Sir faltoo me itna lumba kheech Diya 🙏 2 ko √2 square likh dete tho direct 1/a^2+x^2 ka formula lag jata 🙏

[@IntegralsForYou]

Hi! When I do my videos I try to not use formulas. In this case, if you are allowed to use the integral of 1/(a^2+x^2) formula I'm ok with it, but if you are not allowed to use it or you just want more details about how to solve this integral without using the formula then you have the steps in the video 😉