Integral of x*arccos(x) (by parts + substitution)

[@IntegralsForYou]

👋 Follow @integralsforyou for a daily integral 😉 📸 instagram.com/integralsforyou/ 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐭𝐢𝐨𝐧 𝐦𝐞𝐭𝐡𝐨𝐝𝐬 𝐩𝐥𝐚𝐲𝐥𝐢𝐬𝐭 ► Integration by parts kzbin.info/aero/PLpfQkODxXi4-GdH-W7YvTuKmK_mFNxW_h ► Integration by substitution kzbin.info/aero/PLpfQkODxXi4-7Nc5OlXc0zs81dgwnQQc4 ► Integration by trig substitution kzbin.info/aero/PLpfQkODxXi49OUGvTetsTW61kUs6wHnzT ► Integration by Weierstrass substitution kzbin.info/aero/PLpfQkODxXi4-8kbKt63rs1xwo6e3yAage ► Integration by partial fraction decomposition kzbin.info/aero/PLpfQkODxXi4-9Ts0IMGxzI5ssWNBT9aXJ 𝐅𝐨𝐥𝐥𝐨𝐰 𝐈𝐧𝐭𝐞𝐠𝐫𝐚𝐥𝐬 𝐅𝐨𝐫𝐘𝐨𝐮 ▶️ KZbin kzbin.info 📸 Instagram instagram.com/integralsforyou/ 👍 Facebook facebook.com/IntegralsForYou 𝐃𝐨𝐧𝐚𝐭𝐞 🙋‍♂️ Patreon www.patreon.com/integralsforyou

[@aaaaaaaaaamamabdkzkan]

thank you so much this helped me a lot :) But could you write bigger next time if possible

[@IntegralsForYou]

I'll try! You can watch my last videos, they have more quality, let me know what you think 😉

[@aaaaaaaaaamamabdkzkan]

@@IntegralsForYou i didn't realize this video was 4 years old lol. I will watch them. Thanks for the positive feedback :)

[@IntegralsForYou]

@Duru Thanks for your comments! 💪💪

[@wyl3638]

I wonder why sqrt(sin²(u))＝sinu instead of ±sinu

[@IntegralsForYou]

Hi! I found this article that explains it: math.stackexchange.com/questions/1118400/trig-substitution-why-can-we-ignore-the-absolute-value 😉

[@wyl3638]

@@IntegralsForYou thanks a lot😘

[@IntegralsForYou]

@@wyl3638 My pleasure! ☺

[@halilaa2116]

hi ı have one question why we say x=cos(u) instead of x=sin(u) ?? I was solution with x=sin(u) and answer is =x^2/2*arccosx+1/4(arcsinx-xsqrt(1-x^2)+C These answers are same?

[@IntegralsForYou]

Hi, Ibrahim, how are you? Since: Derivative of arcsin(x) = 1/sqrt(1-x^2) Derivative of arccos(x) = -1/sqrt(1-x^2) Then both next expressions have the same derivative: (x^2/2)arccos(x) - (1/4)arccos(x) - x*sqrt(1-x^2) + C (x^2/2)arccos(x) + (1/4)arcsin(x) - x*sqrt(1-x^2) + C Then both answers are the same and they are the solution of the integral of x*arccos(x). In this video I chose x=cos(u) in order to have the solution in terms of arccos(x).

[@godfran24]

Nice ;)

[@IntegralsForYou]

:-D

[@locopute1098]

at 3:49 where did the 2 infront of cos come from?

[@IntegralsForYou]

Hi! Since the integral of f'(x)cos(f(x)) dx = sin(f(x)) , where f'(x) is the derivative of f(x), in our case we have f(u) = 2u and its derivative is f'(u)=2. We need to multiply cos(2u) by 2 in order to have its derivative: Integral of cos(2u) du = = (1/2)Integral of 2*cos(2u) du = = (1/2)sin(2u) If you don't like this method, you can use substitution: Integral of cos(2x) dx = kzbin.info/www/bejne/faW7iqyApth7l80

[@jmause5775]

why sin(arccos(x))=square(1-x2) and cos(arccos(x))=x?

[@IntegralsForYou]

Hi jj ms! 1. cos(arccos(x))=x By definition, if cos(x)=y then x=arccos(y). So, x=arccos(y)=arccos(cos(x)). 2. sin(arccos(x))=sqrt(1-x^2) We can use the formula 1=sin^2(y)+cos^2(y) ==> sin(y)=sqrt(1-cos^2(y)). Let y=arccos(x): sin(arccos(x))=sqrt(1-cos^2(arccos(x)))=sqrt(1-x^2) Hope it helped!

[@jmause5775]

Integrals ForYou Thanks, you saved my life. Let's see if I pass the calculus exam next week

[@IntegralsForYou]

I wish you the best for your exam, and if you have more questions here I am! ;-D