# Integration Problem **URL:** \ **Category:** Factual Questions\ **Created:** [March 25, 2002, 9:00pm UTC](https://boards.straightdope.com/t/integration-problem/100071 "2002-03-25T21:00:45Z")\ **Posts on this page:** 18\ **Page:** 1
**Author:** ![mcribbit](https://avatars.discourse-cdn.com/v4/letter/m/df788c/32.png) [@mcribbit](https://boards.straightdope.com/u/mcribbit)\ **Post date:** [March 25, 2002, 9:00pm UTC](https://boards.straightdope.com/t/integration-problem/100071/1 "2002-03-25T21:00:45Z")
Please can someone help me to integrate: sin^3x ---
**Author:** ![Clarissa\_xx](https://avatars.discourse-cdn.com/v4/letter/c/dc4da7/32.png) [@Clarissa\_xx](https://boards.straightdope.com/u/Clarissa_xx)\ **Post date:** [March 25, 2002, 9:06pm UTC](https://boards.straightdope.com/t/integration-problem/100071/2 "2002-03-25T21:06:26Z")
INT(sin^3 x)dx= (-1/3)(cos x)(sin^2 x + 2) Oh wait, did you want me to show my work???😃 ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [March 25, 2002, 9:07pm UTC](https://boards.straightdope.com/t/integration-problem/100071/3 "2002-03-25T21:07:58Z")
Write sin^3x as (sin x)_(sin^2x) and use sin^2x + cos^2x = 1 to write this as (sin x)_(1 - cos^2x) which equals sin x - sin x cos^2x Now sin x integrates to -cos x and (by the chain rule) sin x cos^2x integrates to -1/3 cos^3x. So the whole thing integrates to -cos x + 1/3 cos^3x +c ---
**Author:** ![Clarissa\_xx](https://avatars.discourse-cdn.com/v4/letter/c/dc4da7/32.png) [@Clarissa\_xx](https://boards.straightdope.com/u/Clarissa_xx)\ **Post date:** [March 25, 2002, 9:10pm UTC](https://boards.straightdope.com/t/integration-problem/100071/4 "2002-03-25T21:10:36Z")
I always forget the obligatory +C ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [March 25, 2002, 9:13pm UTC](https://boards.straightdope.com/t/integration-problem/100071/5 "2002-03-25T21:13:03Z")
That’s a mark off where I come from 🙂 ---
**Author:** ![Clarissa\_xx](https://avatars.discourse-cdn.com/v4/letter/c/dc4da7/32.png) [@Clarissa\_xx](https://boards.straightdope.com/u/Clarissa_xx)\ **Post date:** [March 25, 2002, 9:18pm UTC](https://boards.straightdope.com/t/integration-problem/100071/6 "2002-03-25T21:18:02Z")
Yea, here too!! 😉 ---
**Author:** ![mcribbit](https://avatars.discourse-cdn.com/v4/letter/m/df788c/32.png) [@mcribbit](https://boards.straightdope.com/u/mcribbit)\ **Post date:** [March 25, 2002, 9:18pm UTC](https://boards.straightdope.com/t/integration-problem/100071/7 "2002-03-25T21:18:30Z")
> [@](#): > > sin x cos^2x integrates to -1/3 cos^3x Sorry to be stupid but i don’t get this bit. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [March 25, 2002, 9:24pm UTC](https://boards.straightdope.com/t/integration-problem/100071/8 "2002-03-25T21:24:38Z")
> [@](#): > > \*Originally posted by mcribbit \* > \*\* > > Sorry to be stupid but i don’t get this bit. \*\* Differentiate cos^3x using the chain rule. In full y = cos^3x = (cos x)^3 = u^3 where u = cos x Then dy/du = 3u^2 and du/dx = -sin x By the chain rule, dy/dx = dy/du \* du/dx = -3u^2 sin x = -3sin x (cos x)^2 i.e. d/dx {cos^3 x} = -3sin x cos^2 x Integrate both sides: INT {-3sin x cos^2x} = cos^3x Divide by -3: INT {sinx cos ^2x} = -1/3 cos^3x This splitting up the integrand is a standard trick and will work for any odd power of sin x or cos x. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [March 26, 2002, 3:09am UTC](https://boards.straightdope.com/t/integration-problem/100071/9 "2002-03-26T03:09:20Z")
According to the _Standard Mathematical Tables_ from the Chemical Rubber Publishing Company the integral of sin[sup]3[/sup] x = -1/3_cos x_(sin[sup]2[/sup] x + 2) + const. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [March 26, 2002, 3:49am UTC](https://boards.straightdope.com/t/integration-problem/100071/10 "2002-03-26T03:49:51Z")
Y’know, I would feel a lot better about this thread if I could think of any possible reason, other than a homework assignment, why the OP would want to know this. ---
**Author:** ![mcribbit](https://avatars.discourse-cdn.com/v4/letter/m/df788c/32.png) [@mcribbit](https://boards.straightdope.com/u/mcribbit)\ **Post date:** [March 26, 2002, 8:13am UTC](https://boards.straightdope.com/t/integration-problem/100071/11 "2002-03-26T08:13:09Z")
Yeah, chronos you were right, it is for homework. Unfortunatley the integrating skills that we have been taught up to date have not allowed us to handle this sort of integrating. I have no idea why I got asked this question in the homework! ---
**Author:** ![Clarissa\_xx](https://avatars.discourse-cdn.com/v4/letter/c/dc4da7/32.png) [@Clarissa\_xx](https://boards.straightdope.com/u/Clarissa_xx)\ **Post date:** [March 26, 2002, 2:50pm UTC](https://boards.straightdope.com/t/integration-problem/100071/12 "2002-03-26T14:50:33Z")
David, where do you think I got my answer??? ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [March 27, 2002, 1:00am UTC](https://boards.straightdope.com/t/integration-problem/100071/13 "2002-03-27T01:00:09Z")
> [@](#): > > \*Originally posted by Clarissa\_xx \* > \*\*David, where do you think I got my answer??? \*\* Ah, but his had the +const. 🙂 ---
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [March 27, 2002, 2:15am UTC](https://boards.straightdope.com/t/integration-problem/100071/14 "2002-03-27T02:15:37Z")
I guess that means **Clarissa** ’s using the Chemical Rubber Publishing Company’s _Nonstandard Mathematical Tables_. If you’ve a keen eye, you can differentiate between them. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [March 27, 2002, 3:17am UTC](https://boards.straightdope.com/t/integration-problem/100071/15 "2002-03-27T03:17:17Z")
> [@](#): > > \*Originally posted by Clarissa\_xx \* > \*\*David, where do you think I got my answer??? \*\* I didn’t notice that you had already posted the correct answer. I had no intention of working the problem the hard way, so I started the post then went looking for my book. Clever eh? By the time I found it you beat me to it. ---
**Author:** ![Johanna](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/johanna/32/8318_2.png) [@Johanna](https://boards.straightdope.com/u/Johanna)\ **Post date:** [March 27, 2002, 12:58pm UTC](https://boards.straightdope.com/t/integration-problem/100071/16 "2002-03-27T12:58:29Z")
I opened this thread all set to discuss multiracial neighborhoods. ---
**Author:** ![Clarissa\_xx](https://avatars.discourse-cdn.com/v4/letter/c/dc4da7/32.png) [@Clarissa\_xx](https://boards.straightdope.com/u/Clarissa_xx)\ **Post date:** [March 29, 2002, 3:14pm UTC](https://boards.straightdope.com/t/integration-problem/100071/17 "2002-03-29T15:14:18Z")
Zut has it! ‘Nonstandard’ indeed! 😉 ---
**Author:** ![sailor](https://avatars.discourse-cdn.com/v4/letter/s/a587f6/32.png) [@sailor](https://boards.straightdope.com/u/sailor)\ **Post date:** [March 29, 2002, 4:57pm UTC](https://boards.straightdope.com/t/integration-problem/100071/18 "2002-03-29T16:57:03Z")
\>\> If you’ve a keen eye, you can differentiate … We’re trying to _integrate_ which is pretty much the opposite. 😉