integral of ln(sqrt(x+1)+sqrt(x)) by trig sub

We will use trigonometric substitution to intgrate ln(sqrt(x+1)+sqrt(x)). This is a great integration excercise for calculus 2 students. For more calculus tutorials, check out my new channel ‪@bprpcalculusbasics‬ 👉 / justcalculus
integral of sec^x, standard result, • integral of sec(x) ,
integral of sec^x, 4 ways, • integral of sec(x), 4 ...
integral of sec^3x, • Integral of sec^3x, ca...
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[@SatyaVenugopal]

If you put this in your Calc 2 final and your students watch your videos... Ohhhh, I see what you're doing there! XD

[@Risu0chan]

Noticing that √x and √(1+x) are close to sinh and √(1+sinh²) = cosh, I made the substitution x = sinh²(t). The logarithm simplifies nicely log(sinh(t) + cosh(t)) = log(exp(t)) = t

[@margintoosmall925]

Very fascinating video! I was able to solve this with integration by parts, but your solution is much more elegant and less tedious.

[@wristdisabledwriter2893]

I'd luv to see you try to differentiate the amswer

[@Bermatematika]

Thank you for the beautiful problem and good solution. It took me some time to be able to solve this :). My approach is a little different . I will make a video about it when I have time :).

[@zygoloid]

For another approach with no trig sub, you can apply integration by parts directly to the initial integral:

[@PackSciences]

Why did you buy a giant toblerone? Next video on triangular prism geometry?

[@jadewolf3416]

Great video! What classes do you teach tho? Since iirc Ur in cali

[@wedemeyermath4896]

Beautiful problem. Thoroughly enjoyed working through this one.

[@adandap]

I think there's an easier (less algebra) way. Put x = [sinh(t)]^2 so that 1+x = [cosh(t)]^2 and use sinh(t) = 1/2 ( exp(t)-exp(-t) ) etc, so the argument of the log just becomes exp(t), and you get Integral (2t . sinh(t). cosh(t)) etc.

[@jnbaratech2122]

Awesome! I am in love with you now xD greetings from Spain

[@sergioh5515]

Ik I'm not the only one who replayed the intro a few times 😇🤣😲

[@Jeff-wc5ho]

LEGENDARY INTRO

[@marcioamaral7511]

This was great!

[@rishijha2583]

your explanation is best but you can do it by parts considering log'''''' as first function and dx as second one

[@GurkiratSingh-ds8dq]

You know, BPRP, just a suggestion.

[@soliscrown1272]

Very cool vid. That's a good test question.

[@Magic73805]

Thank you very much sir

[@willbishop1355]

Great problem. I got stumped by the sec^3 integral, need to watch your old video!

[@sumitshingare7484]

can we apply by parts on the original expression instead of substitution.