Improper integral with two infinite bounds | AP Calculus BC | Khan Academy

Infinite both ways! Created by Sal Khan.

AP Calculus BC on Khan Academy: Learn AP Calculus BC - everything from AP Calculus AB plus a few extra goodies, such as Taylor series, to prepare you for the AP Test

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[Charlie-hpug]

why would anyone ever down-vote a single khan academy video? All he does is help us all...

[junes_gloom]

building off of that, arcsin is defined literally as a function into which you place a ratio between -1 and 1, and it gives you the angle (between -pi/2 and pi/2) for which the sine of the angle is said ratio. (notice the restricted range here is needed as sin(0) = sin(2pi) = sin(4pi) etc.) With this definition, we observe the composition of the two functions, asin(sin(x)) and translate it to literally, the angle given by the ratio given by the sine of the angle, x, or simply x.

[AnalCrumb]

Just going through calc 2 with your last handful of videos? Works for me!

[anelemadonda]

i really didn't understand how the the tangent came from but i do get the intuition of how does this go this go along thanks Khan academy. Sal you are the hero

[burlapsack]

With a graph like this, why can't you have just taken half of it then just added it to itself? You know that it's a mirrored graph along the y axis.

[vidhit]

where did he get 5tan theta from ???? please help 

[williambyrd]

The arcsine function is defined as the function that will take any number, and tell you what angle that you would need to take the sine of to get that number.
that can be written mathematically as:
If arcsin(y)=x , then sin(x) = y
Now lets look at the second equation, and take the arcsine of both sides.
So, arcsin(sin(x)) = arcsin(y)
But the first equation told us that arcsin(y) = x
We can substitute x in for arcsin(y)
So, arcsin(sin(x)) = x

[jfjiazjzjzjzjz]

Probably. It looks symmetrical, and it seems easy to prove. But without that proof defined, then you can't assume that they are symmetrical.

[uimasterskill]

Hey Sal, great video and very well explained concept! I actually understood almost everything, except for one thing: how do you prove that if x=tan(theta), than arctan(x)=theta? Essentially, this proof comes down to proving that arcsin(sin(x))=x, or sin^(-1)(sin(x))=x, which I just can't seem to wrap my head around.
So, to sum it up, how do you prove that arcsin(sin(x))=x? And no, the answer "they are inverse functions" won't do it for me. I want mathematical proof. Thank you!

[joshuapeek]

I thought in order to do trig sub the 25+x^2 had to be sqrt'ed?

[justinwalker]

How tan squared theta becomes secant theta

[KrOmEiFiCaTiOn]

Thank you so much. Really helped me understand this. But that's totally pink, not orange...

[JohnSmith-nsni]

Why don't you use inverse trigonometric substitution? I feel that it would be simpler.

[DanielR-MIDI]

couldn't theta be other angles instead of just π/2 and -π/2? Like i get the arctan's domain is limited to -π/2 to π/2, but when x is initially defined, its defined as 5tan(theta), so couldn't theta be any angle that its tangent is undefined?

[Hitori]

@Snail Cumming he is using trig substitution it looks like.

[jeremykrieg]

as function is even, it is symmetrical ....

as f(-x)=f(x)

[Swimdarkenigma]

What if you were using degrees instead of radians?

[liquidred]

sorry why is arctan(-infinity)=-π/2? He just kinda drew a down arrow and was like "look. It's -pi/2 guys"

[neelmodi]

wait why does he split it up into two integrals? cant he just set n-->infinity so the whole integral is from -n to +n?i tried it and got the same answer 50pi

[tempjohn]

I love your videos and I watch them all, but aside from the fact that you are speaking English, I don't understand anything else. I keep hoping that by osmosis I will pick up some new maths, but given that I barely got thru basic college algebra, I am not hopeful.