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Integrate sec^2(x)

davidsmith

6

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?

differentiate tan x.

A t-substitution will work.

Reply 3

Swayum

18

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:

).

Reply 4

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.

Reply 5

Lord of the Flies

19

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.

Reply 6

Zuzuzu

12

I've been trying to do this all day with no success. Any hints? It's beginning to irritate me.

Reply 7

Mr M

20

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.

Reply 8

Zuzuzu

12

Thanks. Of all the methods I tried, I can't believe I didn't see u=secx or u=sec^2x.

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

Reply 10

16Characters....

13

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.

Reply 11

Abhishek238

1

Use integration by parts

Reply 12

username3754758

17

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)

Reply 13

simon0

13

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)).

(edited 5 years ago)

Integrate sec^2(x)

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