davidsmith
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#1
Integrate sec^2(x)

I know the answer is tan(x) but I want to be able to prove it...How do I prove this?
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yusufu
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#2
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differentiate tan x.
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Glutamic Acid
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A t-substitution will work.
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Swayum
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(Original post by Glutamic Acid)
A t-substitution will work.
Um, but then you're using the result that tanx differentiates to sec^2x so it's cheating. That's assuming that we can't use that result (if we can, yusufu's method is better :p:).
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anomalocaris
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I guess there is, however it still uses the fact that (tgx)' = (secx)^2.

I = ∫(secx)^2 * dx
let u = tgx
∴du = (secx)^2 * dx
I = ∫du
=u + c
=tgx + c

Hopefully that helps.
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Lord of the Flies
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(Original post by anomalocaris)
I guess there is, however it still uses the fact that (tgx)' = (secx)^2.

I = ∫(secx)^2 * dx
let u = tgx
∴du = (secx)^2 * dx
I = ∫du
=u + c
=tgx + c

Hopefully that helps.
Pretty sure the OP figured it out in the space of three years.
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Zuzuzu
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I've been trying to do this all day with no success. Any hints? It's beginning to irritate me.
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Mr M
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(Original post by Zuzuzu)
I've been trying to do this all day with no success. Any hints? It's beginning to irritate me.
There is a stupid way that avoids knowing how to differentiate tan x.

You can make a substitution u = sec x.

You will then need to make a second substitution.

You still need to know the Pythagorean identities.
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Zuzuzu
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Thanks. Of all the methods I tried, I can't believe I didn't see u=secx or u=sec^2x.
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BrasenoseAdm
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If anyone searching for a solution to this problem comes across this page, here's how its done.

INT sec(^2)x dx

You have to use a substitution. Let u = tan x. We know tan x = sin x / cos x and using the quotient rule, du/dx = 1/sec^2 x. So rearranging, du = sec(^2)x * dx

Substitute and you have INT du

which is just u + c.

Substitute u = tan x and you have tan x + c
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16Characters....
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(Original post by BrasenoseAdm)
If anyone searching for a solution to this problem comes across this page, here's how its done.

INT sec(^2)x dx

You have to use a substitution. Let u = tan x. We know tan x = sin x / cos x and using the quotient rule, du/dx = 1/sec^2 x. So rearranging, du = sec(^2)x * dx

Substitute and you have INT du

which is just u + c.

Substitute u = tan x and you have tan x + c

If I am reading that correctly, you use that the derivative of tan x = sec^2 x which is itself sufficient proof.
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Abhishek238
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Use integration by parts
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dj_ad_1
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sec^2(x) = 1/cos^2(x)
so use quotient rule with u=1 and v=cos^2(x)
the derivative of cos^2(x) can be worked out using chain rule
Put that in the quotient rule formula
you should get the answer sec(x)tan(x)
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simon0
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(Original post by dj_ad_1)
sec^2(x) = 1/cos^2(x)
so use quotient rule with u=1 and v=cos^2(x)
the derivative of cos^2(x) can be worked out using chain rule
Put that in the quotient rule formula
you should get the answer sec(x)tan(x)
First, this is a very old thread.

Second, this discussion is of the INTEGRAL of sec^2(x), not the derivative (and the derivative of sec^2(x) is not sec(x)tan(x)).
Last edited by simon0; 1 month ago
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