Integral of ln(x)/x^3 (by parts)

[@IntegralsForYou]

✍🏼 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

[@haile9209]

Great teacher

[@IntegralsForYou]

Thanks! 😊

[@gelbkehlchen]

Solution: ∫ln(x)/x³*dx = ∫ln(x)*x^(-3)*dx = ----------------------------------------- Solution by partial integration: Partial integration can be derived from the product rule of differential calculus. The product rule of differential calculus states: (u*v)’ = u’*v+u*v’ |-u’*v ⟹ (u*v)’-u’*v = u*v’ |∫() ⟹ ∫u*v’*dx = u*v-∫u’*v*dx ----------------------------------------- = -ln(x)*x^(-2)/2+1/2*∫1/x*x^(-2)*dx = -ln(x)/(2x²)+1/2*∫x^(-3)*dx = -ln(x)/(2x²)-1/2*x^(-2)/2+C = -ln(x)/(2x²)-1/(4x²)+C = -[2*ln(x)+1]/(4x²)+C Test by deriving: {-[2*ln(x)+1]/(4x²)+C}’ = {-2/x*4x²+[2*ln(x)+1]*8x}/(16x^4) = {-8x+[2*ln(x)+1]*8x}/(16x^4) = {-1+2*ln(x)+1}/(2x³) = 2*ln(x)/(2x³) = ln(x)/x³ everything o.k.

[@IntegralsForYou]

Thanks! ❤

[@flor7301]

hi, what method do you use when you do: x^-2/-2 = -1/2x^2

[@IntegralsForYou]

Hi! In this case I am using the next exponential property: a^-n = 1/a^n x^(-2)/(-2) = = (1/(-2))*x^(-2) = = (-1/2)*x^(-2) = = (-1/2)*(1/x^2) = = -1/2x^2 Hope it helped! 💪

[@ulisesjuliosanchez9269]

Thanks

[@IntegralsForYou]

You're welcome! ;-D

[@alfredoperaltal]

Thanks dude!

[@IntegralsForYou]

You're welcome! 😊

[@gisellatorres7931]

Thanks :)

[@IntegralsForYou]

You're welcome! ;-D

[@souly9380]

So helpful ♥️

[@IntegralsForYou]

;-D

[@АйжанАскарова-и3с]

by what method did you decide?

[@IntegralsForYou]

Hi! I did integration by parts 😉

[@silvaasiaineduardo6222]

Tanks

[@IntegralsForYou]

You're welcome! 😊

[@samzo.3239]

Nice

[@IntegralsForYou]

Thanks! 😊

[@zlucky541]

Which book you use?

[@zlucky541]

To get this problems

[@IntegralsForYou]

Hi! I don't use any book, I just do the integrals that people ask me to do. I started with a random book from university but then people started to ask

[@Vanessa-zi4og]

😃

[@Yulinaaguilar591]

una pregunta porque estas restando la x elevada a la 3 ???

[@IntegralsForYou]

Hi, I'm sorry but I don't understand your question, do you mean the use of the formula a^(-n)=1/a^n ?

[@mulongamambwe48]

No volume

[@IntegralsForYou]

Yes, no volume 😊I know that it would be easier for you if I talked but I prefer to let you think why am I doing each step and if you don't understand it you can ask me in a comment. In my opinion, it is the best way to practice integration 😊