#Definite integration
swift muskBOT
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oblique sigil
If not induction, please help with whatever could be done
vivid dirge
Induction does seem to work. Though, the way I did it I needed to do induction on two different kinds of integrals.
oblique sigil
Induction does seem to work. Though, the way I did it I needed to do induction o...
how did you proceed with induction on my result?
vivid dirge
how did you proceed with induction on my result?
Take I(n) to be the definite integral of (sin(nx)/sin(x))^2. In the induction step try using the sum formula for sin((n + 1)x).
rugged kestrel
The idea I got was to use the fact that
$\frac{\sin(n\theta)}{\sin\theta}=\sum_{k=0}^{n-1}\cos\big((n-1-2k)\theta\big)$
silver mothBOT
IdoMeth
rugged kestrel
It's not very hard to prove
vivid dirge
$\frac{\sin(n\theta)}{\sin\theta}=\sum_{k=0}^{n-1}\cos\big((n-1-2k)\theta\big)$
Ohh, nice idea!
Though, you do need to carefully collect terms afterwards.
vivid dirge
How to prove this?
Multiply both sides by sin(t), then use the formula for sin(x)cos(y).
oblique sigil
Multiply both sides by sin(t), then use the formula for sin(x)cos(y).
I wanna start with sinnx/sinx
Not compare rhs with lhs
vivid dirge
Well, my approach didn't include this identity, anyway.
oblique sigil
aah, ill try reducing it using by parts as you did
by parts or not, ill try reducing it anyhow ðŸ˜
oblique sigil
Well, my approach didn't include this identity, anyway.
oof
i couldnt get it
can you drop mroe hints?
oblique sigil
Well, can you show what you did?
I finally got it
oblique sigil
Take I(n) to be the definite integral of (sin(nx)/sin(x))^2. In the induction st...
Using your convention, I_n - I_n-1 = int(sin((2n-1)x)/sinx)
If int(sin((2n-1)x)/sinx) = K_n
Then K_n = K_n-1
Hence K_n = K_1 = π/2
Using recursion I_n -I_0 = nπ/2
And I_0=0
Hence I_10 = 10Ï€/2 = 5Ï€
vivid dirge
Using your convention, I_n - I_n-1 = int(sin((2n-1)x)/sinx)
Right, nice! This does look like what I had.
vivid dirge
Hence I_10 = 10Ï€/2 = 5Ï€
Great!
oblique sigil
Great!
Thankyouuu
Ill close this thread now
+close
Did the command change?
+close
lucid jackalBOT
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oblique sigil
+close
acoustic lava
when will the bot start working
If you go back and have the channel list open, if you then hold on this sub channel and it will give you the option to lock/close the channel