Integral of x^2/(1+x^2) two ways

Integral of x^2/(1+x^2), Trig substitution, calculus 2 integrals, trig sub example!

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[Xnichfrytk]

Gonna start using "wouldn't it be nice..." as a first approach to problem solving from now on XD

[renewableonellc]

You have such an inspiring attitude towards the maths. I wish you taught me math 😂

[mikety]

Figured that was the trick you were gonna use. So sweet and simple - really have to get used to thinking and asking the question, "What is the _most efficient_ method I can use for this problem?" when it comes to things like this. +/-1. Easy. Already got you were gonna use that from the caption before the vid started.

And I also can time it, too:
First method - solving starts at 44 s, final boxing at 205 s = solved in 161 s
Second method - solving starts at 272 s, final boxing ends at 356 s = solved in 84 s.

So this method is literally twice as fast.

FWIW, this same method can be used more generally to solve (x^2)/(x^2 + a) for any value of a.

[Borkography]

I like math! I often do some tasks for fun ... and when I can not or do not want to do it, I sit down in front of a computer, turn on YT and your channel ... and for a long time I watch you solve tasks. A great thing!

I am from Poland and in my opinion, we lack such channels as yours!

[Atlas]

I didn't know I was allowed to add and subtract a one that wasn't there. You always surprise me. I like that you move at a good pace, and you already know what to say. I'm currently doing my undergrad in applied mathematics. Integration is really fun cause people use different approaches to answer it. It's fun to compare what others did, share ideas and tips with each other to help each other find shortcuts or finish faster. Keep the videos coming! New Subscriber.
(I'm currently taking calc 2 😍 )

[garyhuntress]

I was today years old when I learned about the theta world

[VG-ebkd]

We can do it by u-substitution, divide numerator and denominator by x^2 and then we can substitute u = 1/x

We will get something which we can deal with using a partial fraction then integrate it and resubstitute to bring it to x-world.

And of course not forgetting the constant XD

[srijanhansda]

I did get this sum in my exam
And that second method is freaking awesome!

[CuriousK]

Thank you for the simplified method of "inserting denominator in numerator". Really helpful for my exam

[陳太和-lj]

I solve this question by method 2 directly.

[ΆγγελοςΔημόπουλος-φδ]

Very nice video. I have a suggestion for a video. Calculate a definite integral by definition!

[sergiosutti]

Who uses the first method to solve this integral is crazy

[rollingsnowball]

You can also do the first part of a u-sub. Let u = 1 + x ^ 2. You can rewrite both the numerator and denominator in terms of u and simplify. It’s essentially the same thing as method 2

[pritivarshney]

Both the methods were really cool
Trig sub - almost everything cancels which is satisfying
Second method - Excellent logic
Thank you sir for such an amazing video.

[hammode]

We can use the long division and it will be easier :)

[wenhanzhou]

Very nice way to think for second method.

[christophermusso]

Says "Wouldn't it be nice...", and I start hearing the Beach Boys song in my head. Great video as always!

[anshumanchoudhary]

It becomes way easier if you add 1 and subtract 1 from he numerator . That way, you get int[1 - 1/(1+x^2)]dx= x- tan^-1x+c.A single step answer !

[divyajyotibose]

Nice solution involving trigonometry in this

[JoshuaHillerup]

So, there seems to be a ratio of thinking and amount of time working that changes throughout the function on the space of calculus. Is there any way we can use calculus to figure out information to tell us what approach to take? This probably has more than just too variables though, and I am a bit worried about where such a function is discontinuous.