The Official Homework Help Thread

[Oblivia]

Here's how you'd integrate when sine or cosine are squared:

sin^2 Θ = (1 - cos 2Θ) / 2

cos^2 Θ = (1 + cos 2Θ) / 2

You can get these by solving the cosine double-angle formula. There's also plenty of material available on the internet about trigonometric identities - the Wiki page might help quite a bit!

 

[Zandy]

Would anyone by chance know how to integrate trigonometric functions that are raised to the power greater than 1 such as sin^2 x or cos^2 x? I figured out how to integrate regular sinx and cosx functions but I don't understand what to do when there is a power greater than 1...

Oblivia posted a method for integrating sin^2 (x) and cos^2(x) that requires a lot less computation, however, an alternative method is to use integration by parts ^^.

Let u = sin (x) and let dv = sin(x) dx. Then du = cos(x) and v = -cos (x). Note that cos^2 (x) = 1 - sin^2 (x) and so:

The same trick can be applied for integrating f(x) = cos^2 (x).

- - -

Integrating higher powers of the sine and cosine functions become incredibly cumbersome but can be accomplished by successively using integration by parts. However, you can also use reduction formulas for higher powers of sine and cosine.

For each n ≥ 2:

You can then just plug in the power n to the formulas above as you need and successively use the formulas for the reduced integral on the right. Those formulas are rather difficult to memorize though, but, if you'd like me to derive those formulas, just let me know ! It's not too complicated, just a bit messy xP.

 

[Drake789]

Here's how you'd integrate when sine or cosine are squared:

sin^2 Θ = (1 - cos 2Θ) / 2

cos^2 Θ = (1 + cos 2Θ) / 2

You can get these by solving the cosine double-angle formula. There's also plenty of material available on the internet about trigonometric identities - the Wiki page might help quite a bit!

Thank you for the website link. I glanced through it and it looks like it has a lot if useful information about trig!

 

[Tessie]

So awesome to see some integration by parts, or calculus II stuff going on in this thread
if somebody really is clueless or needs help on any math problem that is calculus and beyond (that means no pre calc stuff) I have a really awesome and good friend who is currently studying for his doctorate (Ph.D) in mathematics that I can ask

 

[Drake789]

Oblivia posted a method for integrating sin^2 (x) and cos^2(x) that requires a lot less computation, however, an alternative method is to use integration by parts ^^.

Let u = sin (x) and let dv = sin(x) dx. Then du = cos(x) and v = -cos (x). Note that cos^2 (x) = 1 - sin^2 (x) and so:

The same trick can be applied for integrating f(x) = cos^2 (x).

- - -

Integrating higher powers of the sine and cosine functions become incredibly cumbersome but can be accomplished by successively using integration by parts. However, you can also use reduction formulas for higher powers of sine and cosine.

For each n ≥ 2:

You can then just plug in the power n to the formulas above as you need and successively use the formulas for the reduced integral on the right. Those formulas are rather difficult to memorize though, but, if you'd like me to derive those formulas, just let me know ! It's not too complicated, just a bit messy xP.

Oh wow I didn't even think about using integration by parts xD Ahh okay I didn't realize those formulas even existed, I'll try my best to memorize them! Thank you so much for the help, it makes a lot more sense now

 

[Acruoxil]

Drake789: I've always found the method Oblivia mentioned the easiest, and it can be helpful in a lot of other kinds of problems you might have to solve. While using by parts won't make it wrong in any way, I vastly prefer doing it the former way. Just my two cents.

 

[Zandy]

Drake789: I've always found the method Oblivia mentioned the easiest, and it can be helpful in a lot of other kinds of problems you might have to solve. While using by parts won't make it wrong in any way, I vastly prefer doing it the former way. Just my two cents.

It definitely is a lot easier for integrating sin^2(x) and cos^2(x) if one can remember those particular trigonometric identities. I remember when I took calculus I almost always needed a sheet of trigonometric identities because there's a ton that are used pertaining to different problems and I'd often forget them or get them mixed up, haha .

Unfortunately the identities above will not be of much help when n > 2 though. Functions like cos^3(x) can be written as cos(x)[cos^2 (x)] = cos(x)[(1 + cos 2x) / 2] = [cos(x) + cos(x) cos(2x)]/2, but this function will not be pleasant to integrate even if cos(2x) is replaced with yet another trigonometric identity which is even more to memorize ><.

I'm not familiar with any of the identities for sine/cosine raised to powers greater than 2 though, though some may be useful for simplifying integration. That said, any useful identities probably contain lower powers of sine/cosine which would also need to be reduced by other identities, and the nightmare continues xD. Overall, integrating higher powers of the sine/cosine function gets messy and time consuming very quickly even with the reduction formulas above since each application reduces the problem by an order of 2.

 

[Acruoxil]

It definitely is a lot easier for integrating sin^2(x) and cos^2(x) if one can remember those particular trigonometric identities. I remember when I took calculus I almost always needed a sheet of trigonometric identities because there's a ton that are used pertaining to different problems and I'd often forget them or get them mixed up, haha .

Unfortunately the identities above will not be of much help when n > 2 though. Functions like cos^3(x) can be written as cos(x)[cos^2 (x)] = cos(x)[(1 + cos 2x) / 2] = [cos(x) + cos(x) cos(2x)]/2, but this function will not be pleasant to integrate even if cos(2x) is replaced with yet another trigonometric identity which is even more to memorize ><.

I'm not familiar with any of the identities for sine/cosine raised to powers greater than 2 though, though some may be useful for simplifying integration. That said, any useful identities probably contain lower powers of sine/cosine which would also need to be reduced by other identities, and the nightmare continues xD. Overall, integrating higher powers of the sine/cosine function gets messy and time consuming very quickly even with the reduction formulas above since each application reduces the problem by an order of 2.

Ah yes, I do remember doing that as well but I guess I ended up memorizing them really well since I practiced on this stuff so much.

And um yeah when you use that identity you do it in exactly that manner for sin^3x, but for cos^3x you use the sin^2x + cos^2x=1 => cos^2x=1-sin^2x. Then just use that and you'll get your answer. In the case of sin^3x I find using the subsitiution method after using the identity Oblivia mentioned much easier.

And yea, I know what you mean XD when you don't know the identities, it's better to tread lightly and don't make it too messy lol. TO each his, own, really c:

 

[Zandy]

Ah yes, I do remember doing that as well but I guess I ended up memorizing them really well since I practiced on this stuff so much.

And um yeah when you use that identity you do it in exactly that manner for sin^3x, but for cos^3x you use the sin^2x + cos^2x=1 => cos^2x=1-sin^2x. Then just use that and you'll get your answer. In the case of sin^3x I find using the subsitiution method after using the identity Oblivia mentioned much easier.

And yea, I know what you mean XD when you don't know the identities, it's better to tread lightly and don't make it too messy lol. TO each his, own, really c:

For that particular example I meant that it would have been hard to integrate the result if one used the double angle-related identity for cos^2 (x) immediately ^^.

 

[Oblivia]

Ah yes, I do remember doing that as well but I guess I ended up memorizing them really well since I practiced on this stuff so much.

And um yeah when you use that identity you do it in exactly that manner for sin^3x, but for cos^3x you use the sin^2x + cos^2x=1 => cos^2x=1-sin^2x. Then just use that and you'll get your answer. In the case of sin^3x I find using the subsitiution method after using the identity Oblivia mentioned much easier.

And yea, I know what you mean XD when you don't know the identities, it's better to tread lightly and don't make it too messy lol. TO each his, own, really c:

I totally agree. Memorization is a pretty valuable tool when it comes to trigonometric identities since the technique is so much easier to apply to actual problems. There's an identity for any power of sine or cosine which are all freely available via a quick internet search and usually aren't too difficult to memorize; granted memorization is kinda my forte so this is what always worked best for me.

I've personally always found integration by parts to be needlessly complicated when compared to using identities, though both are certainly effective methods. But yeah, to each their own! Part of the beauty of math is that there's normally multiple different approaches to problem solving and people can choose the method that suits their particular learning style. ^_^

 

[Acruoxil]

Oh gosh, I forgot to respond here. Bell Tree doesn't notify for someone quoting you :/

Well since I didn't study organic so well last year, I'm having trouble with some really basic stuff.

- What exactly is chirality of a compound? What makes a structure chiral?

- What are the sigma and pi bonds? How do you identify and distinguish them?

- I'm having trouble comprehending the mechanism of nucleophilic substitution reactions in haloalkanes and haloarenes ;u; can anyone help me out with that?

- Which ones are the ortho, meta and para positions in a cyclic structure? I have a bit of an idea but I still need something assuring to carry forward with what I think they are.

- How would you label the alpha, beta and gamma in the structure of a compound? What exactly are they, and why do you use them? Here's an example of the kind of structure I'm talking about: http://masterorganicchemistrycom.c.presscdn.com/wp-content/uploads/2012/03/1-carbonyl.png

I have a few more nomenclature related questions but I need to revise my stuff once again to come up with queries. I can't ask these questions from my teacher or something because they'll just end up mocking me on how I should've learnt them in lower classes but I had my problems; that's just how teachers are here ; ___;

Any help would be appreciated c: Thanks in advance! Also major thanks to Rei and Zandy for running this thread haha.

Yea anyone wanna help with that?

 

[Sona]

Bump!

 

[Blu-chu]

Hey! I just found out about this, and just like... wow. Thanks to the two people who are running this!
So in my school, they teach Latin. It might be kinda strange, as people might see it as a dead language (as it is, in a way). They have their reasons though which I won't go into the details.

I am assigned to learn three of the Latin declensions, like 1st declension, 2nd declension, ect. I'm absolutely awful at memorizing, and having to learn all three is just giving me a headache. I'll give an example of what I have to memorize:
"1st Declension
Aqua
Nominative: [Singular] Aqua [Plural] Aquae
Genitive: [Singular] Aquae [Plural] Aquārum" and so on.

Anyone have tips as to help learn and remember these?

 

[KarlaKGB]

Yea anyone wanna help with that?

honestly, theres a lot there to explain, and u can just google it. its not difficult stuff

 

[Sona]

honestly, theres a lot there to explain, and u can just google it. its not difficult stuff

From the opening post:
"This is supposed to be a welcoming environment to ask for help. Do NOT comment on someone's skill level on a subject. Be respectful of all users regardless of what they know or do not know."

Please keep this in mind, thanks

 

[Acruoxil]

That's alright; I did google a bit of that stuff but couldn't figure ;u; guess I just don't know how to find things right XD

 

[Sona]

bump!

 

[Llust]

i had to miss my japanese class today and i dont get any of the notes or homework we have. im going over all of my friends notes but im still really confused tbh so i have barely to no knowledge on how to do the homework >.< id appreciate an explanation for each answer

there are two parts to the homework. the first is pretty much just review and the second part is what i missed out on. both of which are copied from an online textbook provided to us

pt1 (asking for an answer check)

translate the following statements into english:​

Q: 日本へ行くために、日本語を勉強しています。
A: i studied japanese before going to japan

Q: 日本へ行くための準備をする。
A: get ready to travel to japan

Q: 友達のためにケーキを焼きましょう。
A: lets bake a cake for our friend

Q: 病気のため、クラスを休んだ。
A: (s)he was absent due to an illness

Q: 6. 古い魚を食べたために、おなかが痛くなった。
A: his/her stomach began to ache after eating the raw fish

Q: はい、いらっしゃるはずです。
A: yes, (s)he's expected

pt2(need assistance on the questions)

choose possible combinations for the sentence​

1.) このにもつを __________ 。
a. とどきました
b. あずかって
c. 置こう
d. おろしてください

2.) __________ はずですよ。
a. やくに立つ
b. 学会に出る
c. だいじょうぶ
d. もう頼みました

3.) __________ もう頼みました
a. めずらしい
b. ゆうめい
c. 元気
d. このはこをくれた

4.) __________ つもりですが。
a. りょこうに出る
b. りっぱ
c. 本だなにしまった

5.) __________ 方がいいかもしれない。
a. そのおかしより、もっとあまいの
b. 急いだ
c. もっと使いやすいの
d. たばこをすわなかった

 

[JellyLu]

Hi ^^;

I need organic help and Japanese help, so thanks in in advance to anyone who helps me!

For organic, I need to memorize ALOT of mechanisms and reactions. Does anyone have any methods that work for them? I'm about to take the toughest exam of the course in a week and I can't get the hang of the reactions at all. I just have to memorize so many mechs like: formation of ethers, formation of epoxides, nucleophilic substitution and additions (?), and SN and E reactions in general. Memorizing general answers aren't helping because I always miss something (like mistaking an elimination reaction for substitution) >< Any tips would be appreciated nwn

For Japanese, I'm in Kanji Drill and have to make a presentation on 2 kanji. One of the parts of the presentation involves recognizing and explaining the radical of each kanji. I chose 花 and 星. I just want to check to see if I got the right radicals because I'm still unsure. For 花 is the radical the radical for grass, 艹? For 星 is the radical the radical for sun, 日? I'm just checking....also any explanation on exactly how to identify a radical would be appreciated...I don't think googling answers is going to help me much when studying for an exam xD

Thanks in advance and sorry for such silly questions!

 

[Mega_Cabbage]

Hey! I just found out about this, and just like... wow. Thanks to the two people who are running this!
So in my school, they teach Latin. It might be kinda strange, as people might see it as a dead language (as it is, in a way). They have their reasons though which I won't go into the details.

I am assigned to learn three of the Latin declensions, like 1st declension, 2nd declension, ect. I'm absolutely awful at memorizing, and having to learn all three is just giving me a headache. I'll give an example of what I have to memorize:
"1st Declension
Aqua
Nominative: [Singular] Aqua [Plural] Aquae
Genitive: [Singular] Aquae [Plural] Aquārum" and so on.

Anyone have tips as to help learn and remember these?

It helps if you sing a song.

1st Declension (to the song Twinkle, Twinkle Little Star)

A, A-E, A-E, A-M
A, A-E, A-R-U-M
I-S, A-S, I-S too
Now the first declension?s through

Second Declension (to the song Jingle Bells)

U-S I. . . O U-M
O I O-R-U. . . -M
I-S O-S I-S is the 2nd declension?masculine!