Comparison Theorem for improper integral, calculus 2 tutorial

Calculus 2 tutorial. Learn and understand the comparison theorem for improper integrals.

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[qmzp]

I like how your recent releases are perfectly in sync with what I'm studying in calc 2, haha. Thanks for the videos!

[johnmin]

You can't just conclude that's old.
*Don't do that.*
Your explanations are the best haha

[himanshnegi]

Sir its my humble request to please checkout the "jee advance" integration sums.

Believe me they are really complicated so u should make videos on them, as most of the sums blow up my mind.

P.S.- sorry for my bad english.

[AlRoderick]

I think he wants us to be sure that we put down the correct inequality right there.

[futfan]

I figured that the denominator was really close to the sqrt((x^2+sqrt2)^2). The sqrt and square cancel out, so you are left with the intgeral of 1/(x^2+sqrt2), which is 1/sqrt(sqrt2)*arctan(x/sqrt(sqrt2)), which definitely converges, so this one probably does too

[arvindkumar]

We can make this inequality stronger by noting that 1/(x**2 + 1) = 1/sqrt(x**4 + 2*(x**2) + 1) > 1/sqrt(x**4 + 3*(x**2) + 2 ) > 1/sqrt(x**4 + 4*(x**2) + 4 ) = 1/(x**2 + 2)

Where ' ** ' is the power function

Now integrating both the extreme sides we find that the required value lies between ( pi/2 , pi/(2*sqrt(2)) - arctan(1/sqrt(2)) / sqrt(2) )

[chaoticoli]

I think it's unnecessary to prove that 1/x^2 >= 1/(sqrt(x^4 +... even terms + positive constant)). 1/sqrt(x^4) = 1/x^2, so adding more positive things to the denominator makes the entire expression smaller.

[pauljackson]

-2 < 1
(-2)^2 < (1)^2
4 < 1
wait...
When you square the numbers you lose any conditional direction.
It's like when you multiply be a negative number you must change the direction;
squaring multiplies the left by -2 and the right by +1, does the greater-than become a less-than or not!

Regardless, I thumbs upped since that still works and can help find out if the integral even converges.
Of course I always thumbs up your videos anyway.
We need more teachers/professors like yourself.

I like using the word (well non-word) un-ir-regardless-ness-ful-ly.
Sure the affixes may not make sense but it sounds cool/stupid.

[snejpu]

You and Peyam uploading exactly at the same time. : )

[Ispeaktrollish]

Could you do the graph of e^(-e^x))? It's pretty interesting!

[MarkMcDaniel]

I'm guessing that chalk costs much less than dry erase markers.

[rah]

Can't you make the stronger statement that definite integral 1/sqrt(x^4 +3x^2 + 2) < 1 ?

[arfanm]

How did you know I’m taking Calc 2 and I have the test which will cover Comparison Theorem coming Monday

[linearealgebra]

Soo low
Can you sometimes do some multivariable Analysis

[bipinmaurya]

I think It's special for jee mains