# Integration question
**URL:** https://boards.straightdope.com/t/integration-question/323807
**Category:** Factual Questions
**Created:** [September 28, 2005, 4:41am UTC](https://boards.straightdope.com/t/integration-question/323807 "2005-09-28T04:41:59Z")
**Posts on this page:** 20
**Page:** 1
### Author:  [@chaoticbear](https://boards.straightdope.com/u/chaoticbear)
#### Post date: [September 28, 2005, 4:41am UTC](https://boards.straightdope.com/t/integration-question/323807/1 "2005-09-28T04:41:59Z")
Our teacher gave us an assignment to integrate 4 horrible functions using Maple.
That’s a gigantic checkmark - done.
We were also told to integrate one of them by hand. Which I did. He gave us advice on which one not to pick. Well, long story short, I tried to do them all, just to prove that I could. Except for one. Which I can’t figure out.
I have to integrate f(x)=sqrt(tan(x)) wrt x.
How far I got with this was integrate by parts, and got some horrible stuff, which I tried integrating by parts again, and didn’t get anywhere useful. I tried some algebraic tricks, like multiplying the top and bottom by the square root, so that… well, I don’t know what I thought would come of it.
What would be the way to start this?
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### Author:  [@TJdude825](https://boards.straightdope.com/u/TJdude825)
#### Post date: [September 28, 2005, 4:53am UTC](https://boards.straightdope.com/t/integration-question/323807/2 "2005-09-28T04:53:08Z")
I wouldn’t know how to do this (you seem to be a bit further ahead in Calculus than I am) but I’m curious: What’s wrt?
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### Author:  [@Kid\_A\_1](https://boards.straightdope.com/u/Kid_A_1)
#### Post date: [September 28, 2005, 4:55am UTC](https://boards.straightdope.com/t/integration-question/323807/3 "2005-09-28T04:55:56Z")
I’ll WAG that it means with respect to.
As in integrating the function with respect to x.
I wish I could help but it’s been a while since I’ve done integration.
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### Author:  [@TJdude825](https://boards.straightdope.com/u/TJdude825)
#### Post date: [September 28, 2005, 5:13am UTC](https://boards.straightdope.com/t/integration-question/323807/4 "2005-09-28T05:13:04Z")
> [@Kid\_A](#):
>
> I’ll WAG that it means with respect to.
>
> As in integrating the function with respect to x.
>
> I wish I could help but it’s been a while since I’ve done integration.
That would make sense. I would have written that this way:
Find d/dx (f(x) if f(x) = sqrt(tan(x))
But maybe that’s just because my teacher doesn’t use the “wrt” notation. Now I know!
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### Author:  [@TJdude825](https://boards.straightdope.com/u/TJdude825)
#### Post date: [September 28, 2005, 5:16am UTC](https://boards.straightdope.com/t/integration-question/323807/5 "2005-09-28T05:16:32Z")
> [@TJdude825](#):
>
> That would make sense. I would have written that this way:
>
> Find d/dx (f(x) if f(x) = sqrt(tan(x))
>
> But maybe that’s just because my teacher doesn’t use the “wrt” notation. Now I know!
I missed a parenthesis.
Find ([sup]d[/sup]/[sub]dx[/sub])[f(x)] if f(x) = sqrt(tan(x))
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### Author:  [@TJdude825](https://boards.straightdope.com/u/TJdude825)
#### Post date: [September 28, 2005, 5:17am UTC](https://boards.straightdope.com/t/integration-question/323807/6 "2005-09-28T05:17:37Z")
> [@TJdude825](#):
>
> I missed a parenthesis.
>
> Find ([sup]d[/sup]/[sub]dx[/sub])[f(x)] if f(x) = sqrt(tan(x))
I mean:
Find f(x) if ([sup]d[/sup]/[sub]dx[/sub])[f(x)] = sqrt(tan(x))
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### Author:  [@chaoticbear](https://boards.straightdope.com/u/chaoticbear)
#### Post date: [September 28, 2005, 5:43am UTC](https://boards.straightdope.com/t/integration-question/323807/7 "2005-09-28T05:43:51Z")
Yup, it means “with respect to”.
Only on the SDMB could you have 4 responses to a calculus question posted at about midnight.
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### Author:  [@Malacandra](https://boards.straightdope.com/u/Malacandra)
#### Post date: [September 28, 2005, 10:26am UTC](https://boards.straightdope.com/t/integration-question/323807/8 "2005-09-28T10:26:11Z")
> [@chaoticbear](#):
>
> Yup, it means “with respect to”.
>
> Only on the SDMB could you have 4 responses to a calculus question posted at about midnight.
Integrating root tan? It’s horrible. I Googled this one some time ago. You may want to do the same.
The method goes like this:
Integrate by substituting v = sqrt(tan(x)). You will end up having to integrate 2v[sup]2[/sup]dv / (1 + v[sup]4[/sup]). I’ll leave getting this far as an exercise for the student. If you’re cool with integrating by substitution then you should get to here after a while.
So what? Well, 2v[sup]2[/sup]dv / (1 + v[sup]4[/sup]) is horrible, but you can make it a bit easier with this identity: 1+v[sup]4[/sup] = (v[sup]2[/sup]-sqrt(2)v+1)(v[sup]2[/sup]+sqrt(2)v+1)
Now break it down into partial fractions and see how you get on.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [September 28, 2005, 12:24pm UTC](https://boards.straightdope.com/t/integration-question/323807/9 "2005-09-28T12:24:22Z")
Functions like this are why Maple/Mathematica/etc. exist in the first place.
> [@TJdude825](#):
>
> I mean:
>
> Find f(x) if ([sup]d[/sup]/[sub]dx[/sub])[f(x)] = sqrt(tan(x))
But he’s not looking for the derivative. He’s looking for the integral.
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### Author:  [@Malacandra](https://boards.straightdope.com/u/Malacandra)
#### Post date: [September 28, 2005, 12:35pm UTC](https://boards.straightdope.com/t/integration-question/323807/10 "2005-09-28T12:35:28Z")
> [@ultrafilter](#):
>
> Functions like this are why Maple/Mathematica/etc. exist in the first place.
>
> But he’s not looking for the derivative. He’s looking for the integral.
Yes. Differentiating it is a piece of piss, it’s a simple function-of-a-function.
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### Author:  [@Mathochist](https://boards.straightdope.com/u/Mathochist)
#### Post date: [September 28, 2005, 2:05pm UTC](https://boards.straightdope.com/t/integration-question/323807/11 "2005-09-28T14:05:04Z")
> [@ultrafilter](#):
>
> Functions like this are why Maple/Mathematica/etc. exist in the first place.
>
> But he’s not looking for the derivative. He’s looking for the integral.
Which is why **TJ** asked for the function _whose derivative is_ root tan.
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### Author:  [@chaoticbear](https://boards.straightdope.com/u/chaoticbear)
#### Post date: [September 28, 2005, 3:12pm UTC](https://boards.straightdope.com/t/integration-question/323807/12 "2005-09-28T15:12:33Z")
Didn’t think of u-substitution because I had nothing to substitute out for the dx. You should see the two pages of work I have devoted to this so far, and still no solution (I did get close last night, and got almost the right answer, but I have (1+sqrt(tan(x)) instead of 2 somewhere.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [September 28, 2005, 4:26pm UTC](https://boards.straightdope.com/t/integration-question/323807/13 "2005-09-28T16:26:08Z")
This one’s easy. Use the method of guess-and-check. Guess a function, differentiate it, and see if you get the original function back. If you do, you’re done. If not, guess again. The key to the method of guess and check is to make good guesses. Here, I would recommend guessing 1/2\*tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)_cos(x)+sin(x)) . Sure enough, when we check d/dx [1/2_tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)\*cos(x)+sin(x))] , we get sqrt(tan(x)). Simple!
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### Author:  [@Happy\_Fun\_Ball](https://boards.straightdope.com/u/Happy_Fun_Ball)
#### Post date: [September 28, 2005, 6:31pm UTC](https://boards.straightdope.com/t/integration-question/323807/14 "2005-09-28T18:31:43Z")
> [@Chronos](#):
>
> This one’s easy. Use the method of guess-and-check. Guess a function, differentiate it, and see if you get the original function back. If you do, you’re done. If not, guess again. The key to the method of guess and check is to make good guesses. Here, I would recommend guessing 1/2\*tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)_cos(x)+sin(x)) . Sure enough, when we check d/dx [1/2_tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)\*cos(x)+sin(x))] , we get sqrt(tan(x)). Simple!
Such a funny guy!
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### Author:  [@Malacandra](https://boards.straightdope.com/u/Malacandra)
#### Post date: [September 28, 2005, 9:18pm UTC](https://boards.straightdope.com/t/integration-question/323807/15 "2005-09-28T21:18:37Z")
> [@Chronos](#):
>
> This one’s easy. Use the method of guess-and-check. Guess a function, differentiate it, and see if you get the original function back. If you do, you’re done. If not, guess again. The key to the method of guess and check is to make good guesses. Here, I would recommend guessing 1/2\*tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)_cos(x)+sin(x)) . Sure enough, when we check d/dx [1/2_tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)\*cos(x)+sin(x))] , we get sqrt(tan(x)). Simple!
ROFLMBUFAO! 😃😃😃
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### Author:  [@TJdude825](https://boards.straightdope.com/u/TJdude825)
#### Post date: [September 28, 2005, 11:24pm UTC](https://boards.straightdope.com/t/integration-question/323807/16 "2005-09-28T23:24:28Z")
> [@Chronos](#):
>
> This one’s easy. Use the method of guess-and-check. Guess a function, differentiate it, and see if you get the original function back. If you do, you’re done. If not, guess again. The key to the method of guess and check is to make good guesses. Here, I would recommend guessing 1/2\*tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)_cos(x)+sin(x)) . Sure enough, when we check d/dx [1/2_tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)\*cos(x)+sin(x))] , we get sqrt(tan(x)). Simple!
I’m impressed. Say, what would you _guess_ this week’s winning lottery numbers are? I’m just wondering.
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [September 29, 2005, 5:14am UTC](https://boards.straightdope.com/t/integration-question/323807/17 "2005-09-29T05:14:32Z")
> [@](#):
>
> I’m impressed. Say, what would you guess this week’s winning lottery numbers are? I’m just wondering.
That one’s even easier to guess, it just takes a little longer. Can I get back to you next week?
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### Author:  [@Malacandra](https://boards.straightdope.com/u/Malacandra)
#### Post date: [September 29, 2005, 10:22am UTC](https://boards.straightdope.com/t/integration-question/323807/18 "2005-09-29T10:22:52Z")
> [@chaoticbear](#):
>
> Didn’t think of u-substitution because I had nothing to substitute out for the dx. You should see the two pages of work I have devoted to this so far, and still no solution (I did get close last night, and got almost the right answer, but I have (1+sqrt(tan(x)) instead of 2 somewhere.
Well the obvious substitutions to try are v = tan x and v = sqrt(tan x), and that’s what I started out with. Unfortunately I wound up with v[sup]2[/sup] dv / (1 + v[sup]4[/sup]) and couldn’t see how to make any progress. It was only when I Googled that I found out about the partial fraction (I wouldn’t have figured out how to factorise the denominator as a pair of trinomials in a month of wet Sundays).
Fortunately I don’t depend upon advanced integration to make a living!
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### Author:  [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)
#### Post date: [September 29, 2005, 2:20pm UTC](https://boards.straightdope.com/t/integration-question/323807/19 "2005-09-29T14:20:18Z")
> [@Malacandra](#):
>
> Fortunately I don’t depend upon advanced integration to make a living!
If you did though, you could just go [here](http://integrals.wolfram.com/) and enter “Sqrt[Tan]” into the magic box. Career saved!
Or career made redundant, one of the two.
I notice **Chronos’s** solution appears different from the one given by the website, but it might be equivalent anyway. I haven’t taken the time to check yet.
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### Author:  [@Pleonast](https://boards.straightdope.com/u/Pleonast)
#### Post date: [September 29, 2005, 3:19pm UTC](https://boards.straightdope.com/t/integration-question/323807/20 "2005-09-29T15:19:18Z")
> [@Chronos](#):
>
> This one’s easy. Use the method of guess-and-check. Guess a function, differentiate it, and see if you get the original function back. If you do, you’re done. If not, guess again. The key to the method of guess and check is to make good guesses. Here, I would recommend guessing 1/2\*tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)_cos(x)+sin(x)) . Sure enough, when we check d/dx [1/2_tan(x)^(1/2)\*cos(x)\*2^(1/2)\*arccos(cos(x)-sin(x))/(cos(x)_sin(x))^(1/2)-1/2_2^(1/2)\*ln(cos(x)+2^(1/2)\*tan(x)^(1/2)\*cos(x)+sin(x))] , we get sqrt(tan(x)). Simple!
Ah, you have physicist training, I see. 🙂 I found I could get wonderful guesses with prayer and Bible study. And by Bible, I mean the one written by the Prophets Gradshteyn and Ryzhik.
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