Further maths: Further Calculus

Risermax

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

Reply 1

Risermax

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).

Reply 2

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?

Reply 3

mqb2766

Original post by Risermax

Its also worth noting that it looks like youre asking about question 2 (solution) but have posted question 3.

Reply 4

Risermax

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

Risermax

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?

Reply 6

Interea

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.)

Reply 7

Arconik

Original post by Risermax

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

Reply 8

mqb2766

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.

Reply 9

mqb2766

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 11 months ago)

Reply 10

Risermax

Original post by mqb2766

It would help to post question 2.

Reply 11

mqb2766

Original post by Risermax

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

Risermax

Original post by mqb2766

oh ok I understand now thanks.

Reply 13

mqb2766

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.