The Student Room Group

Substitution for Definite Integrals with Trig

20150207_182537.jpg

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

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

Original post by Airess3
20150207_182537.jpg

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

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

sec^2 θ
Original post by Airess3
20150207_182537.jpg

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)
Yes, pretty much. You get that:

dx=sec2(θ)dθ \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



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

Original post by Airess3
sec^2 θ
(edited 9 years ago)
Reply 5
Original post by Airess3
20150207_182537.jpg

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
Original post by TeeEm
you may find some of these examples useful

PDF.pdf


Thanks! :smile: But can you explain how you got from the step before to 1/2 + 1/2 cos2θ in the third line in example 3? Is it derived from a certain trig identity?
Reply 7
Original post by Airess3
Thanks! :smile: But can you explain how you got from the step before to 1/2 + 1/2 cos2θ 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
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


Photo on 08-02-2015 at 16.59.jpg Photo on 08-02-2015 at 17.01.jpg

So I got the answer to be 1? Is that the right answer or did something go wrong?
Reply 9
Original post by Airess3
Photo on 08-02-2015 at 16.59.jpg Photo on 08-02-2015 at 17.01.jpg

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
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.
(edited 9 years ago)
Reply 11
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 integrates to θ, with limits in θ
Original post by TeeEm
the first 4 lines are correct then it is wrong

1 integrates to θ, with limits in θ


Ah, ok, so the final answer is pi/4.

Quick Reply

Latest