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Further maths: Further Calculus

Can someone help me with the question below this? I don’t understand how 3x^2+4 changes too 4sec^2(theta).
Reply 1
Original post by Risermax
Can someone help me with the question below this? I don’t understand how 3x^2+4 changes too 4sec^2(theta).

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What substitution are you using? (I'm guessing that's set up earlier in the solution to allow finding values for theta, and involves tan.)

What trig identity do you know involving tan and sec?
Original post by Risermax
7B085DDD-7529-47A1-84CD-84A0AF322457.jpeg
60609211-F053-45F7-BEE7-A5323DD96858.jpeg

Its also worth noting that it looks like youre asking about question 2 (solution) but have posted question 3.
Reply 4
Original post by mqb2766
Its also worth noting that it looks like youre asking about question 2 (solution) but have posted question 3.

yh its question 2 my bad.
Reply 5
Original post by Interea
What substitution are you using? (I'm guessing that's set up earlier in the solution to allow finding values for theta, and involves tan.)

What trig identity do you know involving tan and sec?

I know that tan^2(theta) = sec^2(theta) -1. Am I meant to say x = that?
Original post by Risermax
I know that tan^2(theta) = sec^2(theta) -1. Am I meant to say x = that?

That identity will be useful after you've substituted for x. (You seem to have cut off the start of the solution on the previous page, in order to find the theta values for each x you should already have x = something.)
Original post by Risermax
7B085DDD-7529-47A1-84CD-84A0AF322457.jpeg
60609211-F053-45F7-BEE7-A5323DD96858.jpeg


Can't answer your question but did try Q3 for revision. Can you verify for me if that answer is (pi.root3)/9?
Original post by Arconik
Can't answer your question but did try Q3 for revision. Can you verify for me if that answer is (pi.root3)/9?


Yup, thats correct.
Original post by Risermax
I know that tan^2(theta) = sec^2(theta) -1. Am I meant to say x = that?


It would help to post question 2.
(edited 1 year ago)
Reply 10
Original post by mqb2766
It would help to post question 2.

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Original post by Risermax
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As this question is something like
(1 + x^2)^(1/2)
doing x = tan(u) or x = sech(u) are obvious things to try. Here you have
(4 + 3x^2)^(1/2)
so as they suggest
x = 2/sqrt(3) tan(theta)
Then
4 + 3x^2 = 4 + 4tan^2(theta) = 4sec^2(theta)
Reply 12
Original post by mqb2766
As this question is something like
(1 + x^2)^(1/2)
doing x = tan(u) or x = sech(u) are obvious things to try. Here you have
(4 + 3x^2)^(1/2)
so as they suggest
x = 2/sqrt(3) tan(theta)
Then
4 + 3x^2 = 4 + 4tan^2(theta) = 4sec^2(theta)

oh ok I understand now thanks.
Original post by Risermax
oh ok I understand now thanks.


Its a reasonably common integration question where you have something like
y = (1 +/- x^2)^(1/2)
squaring both sides and rearranging, its simply
+/- x^2 + y^2 = 1
So using a trig or hyperbolic substitution to transform that expression shouldn't be surprising as its essentially pythagoras (trig identities).

In
https://properhoc.com/maths-and-logic/calculus/integrals/the-integral-of-sqrt1-x2-dx/
there is the usual algebraic slog is at the start, but "using geometry" section shows how/why the trig function transformation is necessary. So the integral of
(1 - x^2)^(1/2)
is the area under a circle which is equal to
sector + triangle
The area of the sector involves an arc trig function as it maps the integration x-limits to angle.
(edited 1 year ago)

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