Samagra G.

asked • 08/01/16

integration

We have to find the integration of ∫20160  [arctanx]dx the limits are from 0 to 2016 . 
[.] means greastest integer

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Kenneth S. answered • 08/01/16

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Expert Help in Algebra/Trig/(Pre)calculus to Guarantee Success in 2018

Michael W.

Deanna, that won't work here, because we're not integrating inverse trig...we're integrating the greatest integer function.
 
I thiiink my comment below, in response to Michael J's answer, should give Samagra a place to start.
 
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08/01/16

Michael J. answered • 08/01/16

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Mastery of Limits, Derivatives, and Integration Techniques

Mark M.

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∫u-1du = lnlul+C.  Also, tan-1x is the arctangent function, not cotx!!
 
Mark M. (Bayport, NY)
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08/01/16

Michael J.

You caught my mistake, I ended up doing the derivative rather than integral.  Since you corrected my mistake, you would you to elaborate on your part?
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08/01/16

Samagra G.

Someone please help me
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08/01/16

Michael W.

Samagra,
 
Sorry to see all of the confusion regarding your question!
 
So, to do this integral, we need to get a sense of what [arctanx] looks like.  The greatest integer function jumps from one value to the next, right?  So, we're going to have to split up the integral where the function jumps, and we'll end up with a series of rectangles.  The area of the rectangles shooooould be pretty easy, so the challenge here is figuring out how wide the rectangles are.
 
Which means we need to figure out what arctan looks like, and then figure out where it jumps from 0 to 1 or 1 to 2, and so on. 
 
So, either you have to graph arctan to remind yourself what it looks like, or you have to think through the different values that arctan can have.  For example:
 
- arctan (0) = 0.  That is, the angle whose tangent is 0 turns out to be 0 radians. 
- arctan (1) = pi/4.  That is, the angle whose tangent is 1 is pi/4 radians.  pi/4 is still less than 1, though, so if we take the greatest integer of that, we'd still get 0.  If you're graphing it, then what we have so far is a horizontal line at y = 0.
 
So far, we've got a whole lot of nuthin, so the integral will be zero...until arctan jumps to a value that is equal to 1.  At that point, we'll actually have a rectangle with some height.  If you can figure out where that is, and then think about what arctan does beyond that value, then I think you'll be on your way.
 
I hope this at least points you in the right direction...
 
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08/01/16

Michael J.

There is no need to give so many down votes.  It does not help anyone to get the correct solution.
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08/04/16

Michael J.

*It does not help anyone when trying to get a correct answer.
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08/04/16

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