How Do You Integrate (sec x)^4?

Thread starter teng125
Start date Jan 2, 2006
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Integrate

In summary, to integrate (sec x)^4, you can use the power reduction formula for secant and rewrite the integral as (1 + tan^2x)^2. A common substitution for (sec x)^4 is u = tan x, which will simplify the integral to (1 + u^2)^2. Yes, you will need to use the power reduction formula for secant and possibly trigonometric identities to simplify the integral before integrating. Integration by parts can also be used for (sec x)^4, choosing u = sec^2x and dv = sec^2x dx, or vice versa. Special cases to consider when integrating (sec x)^4 include being careful when the upper limit of the integral is

Jan 2, 2006

teng125

May i know how to integrate (sec x)^4 ??

the answer is tan x + 1/3 (tan x)^3

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Jan 2, 2006

Pyrrhus

Homework Helper

Try writing your integral as
[tex] \int (\tan^{2} x + 1) \sec^{2} x dx [/tex]

Jan 2, 2006

teng125

okok...thanx

Jan 2, 2006

fargoth

try using [tex]u=tg(x)[/tex]
[tex]du=\frac{dx}{cos^2(x)}[/tex]
and [tex]\frac{1}{cos^2(x)}=1+tg^2(x)=1+u^2[/tex]

Last edited: Jan 2, 2006

FAQ: How Do You Integrate (sec x)^4?

How do I approach integrating (sec x)^4?

To integrate (sec x)^4, you can use the power reduction formula for secant and rewrite the integral as (1 + tan^2x)^2.

What substitution should I use for (sec x)^4?

A common substitution for (sec x)^4 is u = tan x, which will simplify the integral to (1 + u^2)^2.

Do I need to use trigonometric identities to integrate (sec x)^4?

Yes, you will need to use the power reduction formula for secant and possibly trigonometric identities to simplify the integral before integrating.

Can I use integration by parts for (sec x)^4?

Yes, integration by parts can also be used for (sec x)^4. You can choose u = sec^2x and dv = sec^2x dx, or vice versa, and continue from there.

Are there any special cases to consider when integrating (sec x)^4?

Yes, be careful when the upper limit of the integral is close to a singularity (such as x = π/2 or x = -π/2), as the integral may not converge. In these cases, you may need to use a different approach, such as a trigonometric substitution.