What Mistakes Did I Make in These Trigonometric Integral Problems?

[MitsuShai]

I got these two questions wrong and I'm wondering what i did wrong.

Homework Statement [/b]
1. http://i324.photobucket.com/albums/k327/ProtoGirlEXE/q3.jpg

2. http://i324.photobucket.com/albums/k327/ProtoGirlEXE/q4.jpg


The attempt at a solution[/b]
1. http://i324.photobucket.com/albums/k327/ProtoGirlEXE/100_0639.jpg

2.http://i324.photobucket.com/albums/k327/ProtoGirlEXE/100_0643.jpg
(I put t instead of x while I was doing the problem on accident)

 

[jgens]

Your work is really difficult to read from those pages, but from what I can gather, here are some of the mistakes that you made ...

First Problem: You forgot to distribute the 3 across when you expanded 3cos²(2x). Check your work there. Second, I have no idea how you got your expansion for cos³(2x).

Second Problem: Review your trig identities

 

[jgens]

First Problem: cos(2x)cos(4x) =/= cos²(8x). I'm not really sure how you got that in the first place.

Second Problem: Do it again, and this time be very careful about how you distribute things so that you don't end up with extra terms.

 

[MitsuShai]

First Problem: cos(2x)cos(4x) =/= cos²(8x). I'm not really sure how you got that in the first place.

Second Problem: Do it again, and this time be very careful about how you distribute things so that you don't end up with extra terms.

for the second one, is it just (1/3)tan^3 x +c?

for the first one: there was a -cos^3(2x), so I thought I should split it up like cos^2(2x) and cos (2x) then make cos^2(2x)---> -(1/2)+(1/2)cos(4x)(cos(2x)...

 

[jgens]

for the second one, is it just (1/3)tan^3 x +c?

No. Again, be careful about how you distribute things.

for the first one: there was a -cos^3(2x), so I thought I should split it up like cos^2(2x) and cos (2x) then make cos^2(2x)---> -(1/2)+(1/2)cos(4x)(cos(2x)...

Sure, you can split it up like that, and I followed your work through there. However, cos(4x)cos(2x) =/= cos²(8x). I don't know what would make you think that that was the case.

 

[MitsuShai]

No. Again, be careful about how you distribute things.

I tried it with u substitution, instead of distributing, and got (1/3)tan^3 + x + c

Sure, you can split it up like that, and I followed your work through there. However, cos(4x)cos(2x) =/= cos²(8x). I don't know what would make you think that that was the case.

oh yeah duh, so it's just cos(8x)
then I should get (2/8)x-(3/16)sin(2x)+(3/64)cos(4x)+(1/128)cos(16x)+c, right?

 

[jgens]

No. What makes you think that cos(2x)cos(4x) = cos(8x)? They're not equal.

 

[MitsuShai]

No. What makes you think that cos(2x)cos(4x) = cos(8x)? They're not equal.

they're not? so then what can I do, if they won't separate?

 

[MitsuShai]

omg (1/3)tan^3 x + x +c is wrong too :(
this is so frustrating

 

[jgens]

they're not? so then what can I do, if they won't separate?

They're quite clearly not equal. My question is still what makes you think that they should be in the first place? And I would have tackled this problem using integration by parts. It's really easy to derive a reduction formula and that does pretty much all of the work for you.

 

[jgens]

omg (1/3)tan^3 x + x +c is wrong too :(
this is so frustrating

Why don't you show your steps? From the answers that you've been posting, it looks like you've been distributing terms incorrectly.

 

[MitsuShai]

They're quite clearly not equal. My question is still what makes you think that they should be in the first place? And I would have tackled this problem using integration by parts. It's really easy to derive a reduction formula and that does pretty much all of the work for you.

how can you apply integration of parts to sin^6 x, its just one term?

 

[jgens]

how can you apply integration of parts to sin^6 x, its just one term?

You have to manipulate sin⁶(x) using trig identities first.

integral sec^4 x dx
integral (tan^2 x+1)sec^2 x dx
u= tanx du=sec^2 x
integral (u^2 + 1) du
(1/3)tan^3 x + x + c

u =/= x

 

[MitsuShai]

You have to manipulate sin⁶(x) using trig identities first.



u =/= x

now I'm terribly lost

 

[jgens]

Hopefully this will help you with the first problem ...

[tex]\int\sin{(x)}^{2n}dx=\int[1-\cos{(x)}^2]\sin{(x)}^{2(n-1)}dx=\int\sin{(x)}^{2(n-1)}dx-\int\cos{(x)}^2\sin{(x)}^{2(n-1)}dx[/tex]

[tex]\int\cos{(x)}^2\sin{(x)}^{2(n-1)}dx=\frac{\cos{(x)}\sin{(x)}^{2n-1}}{2n-1}+\frac{1}{2n-1}\int\sin{(x)}^{2n}dx[/tex]

Putting these two together, we see that

[tex]\int\sin{(x)}^{2n}dx=\int\sin{(x)}^{2(n-1)}dx-\frac{\cos{(x)}\sin{(x)}^{2n-1}}{2n-1}-\frac{1}{2n-1}\int\sin{(x)}^{2n}dx[/tex]

[tex]\frac{2n}{2n-1}\int\sin{(x)}^{2n}dx=\int\sin{(x)}^{2(n-1)}dx-\frac{\cos{(x)}\sin{(x)}^{2n-1}}{2n-1}[/tex]

[tex]\int\sin{(x)}^{2n}dx=\frac{2n-1}{2n}\left[\int\sin{(x)}^{2(n-1)}dx-\frac{\cos{(x)}\sin{(x)}^{2n-1}}{2n-1}\right][/tex]

Can you see how this applies to the first problem that you're working? What I posted here is a generalization, so you'll need to figure the specifics out.

 

[jgens]

For the second problem, I'll start with what you already have ...

[tex]\int u^2+1du=\frac{u^3}{3}+u+C[/tex]

Since [itex]u=\tan{x}[/itex], it's now just a matter of replacing each [itex]u[/itex] in the equation above with [itex]\tan{x}[/itex]. The last few times, you've haven't been consistent in this last regard and that's what's causing you so much grief.

 

[jgens]

x/8 - 3/16 sin2x + 3/16 ∫ (1 + cos 4x) dx - 1/16 ∫ (1 + cos 4x) cos 2x dx =
x/8 - 3/16 sin 2x + 3/16 x + 3/64 sin 4x - 1/16x + (1/64)sin 4x + C

What happened to the factor of cos(2x)? You can't just ignore it, so yes, your answer is wrong. The approach that I outlined in one of my previous posts will let you evaluate this integral with very little work, so why don't you try to apply that?

∫ sec^4 x dx = ∫ sec^2 x (sec^2 x dx) = ∫ (1 + tan^2 x) sec^2 x dx = tan x + tan^3 x/3 + C
u = tan x. So, ∫ (u^2 + 1) du = u^3/3 + u + C = tan^2 x/3 + tan x + C
****I got that from somewhere, but I think it's wrong because I really think it should be tan^3 not tan^2.

I'm not going to resolve this one for you since you're making such an obvious mistake. Look at what you wrote above carefully and tell me if it makes sense. It doesn't pay off to be sloppy with your work.