integrate sqrt(x+1) over sqrt(x+2)

In this video, I showed how to use varieous integration technique to integrate a rational radical integrand

[hazwi]

at the end, you can get rid of the absolute value on the natural log since the sum of square roots is always positive

[Mehrdad_Basiry-fjrl]

Fantastic question...🎉🎉🎉.
Learn new methods...

[killer_queen]

It's been a while since I've watched one of your videos also sorry I failed your class😅

[surendrakverma]

Thanks Sir for excellent explanation and good handwriting. 😊

[icannotchoose]

This is very well explained! I'd love to see more integral problems like this.

[The_Bad_Ant]

Awesome explain, thank you teacher 💐💙


31/8/2024
SATURDAY
10:52 AM

[zeravam]

I used the sustitution sen^2(theta)=1/(x+2), obtaining: cosec(theta)=sqr(x+2) and cotan(theta)=sqr(x+1). My integral was: I had a equivalent result:

[marcgriselhubert]

When you arrive at sqrt(u^2 - 1) better use u = cosh(t), with t >0

[holyshit]

(x+1)/(x+2) = u^2
This substitution lead us to integrating rational function without any secants
After substitution proposed by me integration by parts will simplify partial fraction decomposition

[enejedhddhd]

I already memorized radical x^2 - a^2 formula😂 i had an ode exam yesterday

[dougaugustine]

Double substitution and a pair of trig identities. Wow.

[joeykraftx]

Nice. if we let ch t=u as 2nd substitution i find the resolution much easier.

[DEYGAMEDU]

Sir I have sent you a derivative question in your mail, please solve that

[jaychirmade]

Can we use the standard formula for √(x^2-a^2)?

[fantasypvp]

The first thing i thought wheb i saw that integral was to immediately put x+1 = tan^2(u), then the top simplifies, the bottom turns to sec and simplifies, so youve got the integral of sin(u) * dx/du du

[trinugroho]

sir, can you explain how can u^2-1 be sec theta?

[Mephisto]

The absolute value inside ln in the final answer is unnecessary.

[flamewings]

To be honest, I don’t know why am I continuing to learning integration math more… I think I’ve found it funny and interesting since my hight school.

So, for last question in the video, we actually can get rid of absolute value inside the natural logarithm, cause inside we have a sum of two square roots, which is always greater than zero, so we don’t need absolute value. But… Is it a mistake to just leave it here?

[JourneyThroughMath]

As far as the plus/minus is concerned, couldnt you ha e factored it out as a constant and then your final answer is +_answer?

[m.h.]

Wolfram Alpha gives a very different result. Looks like your result only works for x > -1, so only for the real numbers. If you want complex solution, your way doesn't work.