Substitution for Definite Integrals with Trig

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username970964
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#1
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Are steps missing between lines 1 and 2? have no idea how the equation changed so drastically.
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WishingChaff
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#2
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The first line is incorrect anyway.

Given  x = \tan(\theta) , what is  dx ?

(Original post by Airess3)
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Are steps missing between lines 1 and 2? have no idea how the equation changed so drastically.
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username970964
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#3
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(Original post by WishingChaff)
The first line is incorrect anyway.

Given  x = \tan(\theta) , what is  dx ?
sec^2 θ
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Anonymous_0749
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(Original post by Airess3)
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Are steps missing between lines 1 and 2? have no idea how the equation changed so drastically.
my working out would be literally:

[arctan(x) ] pi/4 and 0 limits ; you get the formula in the formula book ? well for mei we do (FP2)
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WishingChaff
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#5
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Yes, pretty much. You get that:

 \text{d}x = \sec^2(\theta) \text{d} \theta

Now, you need to substitute this into your integral, along with the correct limits. But, before doing this, do you know a trigonometric identity that might help here?

Spoiler:
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 1+\tan^2(\theta) =  \sec^2(\theta)



Once this is done, what does your integral now simplify too? It should, of course, be an integral of the dummy variable  \theta

(Original post by Airess3)
sec^2 θ
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TeeEm
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(Original post by Airess3)
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Are steps missing between lines 1 and 2? have no idea how the equation changed so drastically.
you may find some of these examples useful

PDF.pdf
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username970964
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(Original post by TeeEm)
you may find some of these examples useful

PDF.pdf
Thanks! But can you explain how you got from the step before to 1/2 + 1/2 cos2θ dθ in the third line in example 3? Is it derived from a certain trig identity?
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TeeEm
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(Original post by Airess3)
Thanks! But can you explain how you got from the step before to 1/2 + 1/2 cos2θ dθ in the third line in example 3? Is it derived from a certain trig identity?
standard identities useful in integration

cos2x = 1/2 +1/2cos(2x)
sin2x = 1/2 -1/2cos(2x)


they are rearrangements of cos(2x) trigonometric identities
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username970964
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#9
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(Original post by TeeEm)
standard identities useful in integration

cos2x = 1/2 +1/2cos(2x)
sin2x = 1/2 -1/2cos(2x)


they are rearrangements of cos(2x) trigonometric identities
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So I got the answer to be 1? Is that the right answer or did something go wrong?
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TeeEm
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(Original post by Airess3)
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So I got the answer to be 1? Is that the right answer or did something go wrong?
I should be able to read it but at the moment I have forgotten how to reflect my head
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username970964
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#11
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(Original post by TeeEm)
I should be able to read it but at the moment I have forgotten how to reflect my head
Ok, here's the pictures, yesterday my phone ran out of charge so I had to use the camera from my laptop.
Attached files
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TeeEm
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#12
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(Original post by Airess3)
Ok, here's the pictures, yesterday my phone ran out of charge so I had to use the camera from my laptop.
the first 4 lines are correct then it is wrong

∫ 1 dθ integrates to θ, with limits in θ
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username970964
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#13
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#13
(Original post by TeeEm)
the first 4 lines are correct then it is wrong

∫ 1 dθ integrates to θ, with limits in θ
Ah, ok, so the final answer is pi/4.
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