This Integral Will Make You Better At Calculus

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Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

#math #brithemathguy #integral

[BriTheMathGuy]

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[sandeshshrestha]

Solving integral of tanx dx: 🙂
Solving integral of sqrt. tanx dx: 🙁

[PunmasterSTP]

Better at calculus? More like "Thank you for schooling us!" I really liked the showcase of these different techniques and how you can break things up with algebra.

[nontth]

I do this integral with out any help. It took me an hour but Im very proud of it.

[BCQM_BCQM]

Just to clarify, the inverse of hyperbolic tangent function is artanh, not arctanh, with "ar" meaning area instead of arc; also, the domain of artanh is (-1, 1), which does not include any of (u+1/u)/√2. Therefore, the result should be arcoth instead of artanh.
Anyway thanks for sharing this problem, it's fun to solve.

[skylardeslypere]

1:42, right here, you can also use a partial fraction decomposition. Write the denominator as follows:
u⁴+1 = u⁴ + 2u² + 1 - 2u² (adding zero)
= (u²+1)² - (√(2) u)²
= [ u² - √(2) u + 1] [ u² + √(2) u + 1]
And now we have written the denominator as the product of two quadratic factors, which we can split using partial fractions. Then we are just integrating a linear term over a quadratic term, which has a fairly standard type of solution involving logarithms and inverse tangents.

[chessthejameswei]

Great intro (or lack thereof)! I like the just jumping right into it and not wasting any time to tackle this monster!

[GroundThing]

I couldn't remember my trigonometric derivatives, outside sin and cos (though honestly I'm not sure why I didn't just do quotient rule), so I tried to make it something more manageable using Euler's formula and after like 10 minutes of work, I managed to circle my way back around to sqrt(tan(x)) = sqrt(tan(x)).

[manucitomx]

I don’t know if I’m better at Calculus now, I know I was very informed.
Thank you for this channel.

[a.syndeed]

You could use the natural log version of the formula instead of the hyperbolic arctangent one, as more people are familiar with that one. That's the one they taught me in high school.

[matrix]

Quite interesting...
I'm just a high school student and I've started learning calculus these days so this type of question are quite challenging for me. But I love challenges 😁
Thanks for sharing such type of question.

[michaellarson]

I did it by just factoring u^4+1 as You get the same answer although it does take a lot more work.

[TheScienceGuy]

God I remember doing this integral a few years ago 🤦‍♂️

[jacr.z.]

I recently learned how to use power series to expand functions like this and get an approximation and honestly it feels liberating to not have to focus on getting an exact function, especially since applied math makes me worry about my future career

[gurjyotsingh]

Integration of 1/x²-a² = (1/2a)ln|(x-a)/(x+a)| +c, ez

[jamirimaj]

What happens when 0 has a value in a function but is undefined in its integral like this one? How can we compute the definite integral from 0 to 1 for example if 0 is undefined in the formula, but 0 to 1 definitely has an area under the curve?

[adityavikramsinha]

You my man, have earned ALL your subscriptions

[aleksandardashich]

On my exam last year I had to do integral
ln(x²+1)e^sqrt(tan(x)) and it split up basically on ln(x²+1) which was not that hard.But for sqrt(tan(x)) it was little bit complicated, I did everything the same till 1:42 when I used partial fractions.I remember it took me very long time to calculate coefficients 'cause I got somehow all of them zero so I tried three times till I finally resolved them, then it took some of work to finish it but it was very challenging, exam took two hours and I was doing this for about 45 minutes.

[parthhooda]

Now differentiate it to prove that it indeed equals √tanx

[cpotisch]

Great technique! I only know it with partial fractions, which is a lot more work although it does work for every nth root of tan.