How to integrate 1/cos(x)

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Learn the steps on how to find the integral of 1/cos(x) dx using the substitution method.

Music by Adrian von Ziegler

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This integral looks deceptively easy when you first see it.

Peter_

Im doing hw in 2 am and just made a cup of tea before watching this video and the music made everything extra chill

slavitotbwse

I'm in love with the music you just used on your video

natashacalva

excellent tutorial and excellent choice of music. Im trying to get through my cal 2 class atm and this helped me immensely.

Chaoddity

ok guys, if we do smt like that 1/cos = cos/cos^2 = d(sinx)/1 - (sinx)^2 = 1/2*ln((1+sinx)/(1-sinx))

АндрейАмелин-уъ

It's simpler.
Just let u=1/cos(x)
du= sin(x)/cos^2(x) dx = u.root(u^2-1)dx
dx = 1/(u.root(u^2-1))
dx/cos(x) = u.du
= du/root(u^2-1)
the integral of which is found to be cosh^-1(u). by letting u = cosh(y)

So the integral of 1/cos(x) is
Cosh^-1(1/cos(x))
which can be shown to be the same as the answer obtained in the video only more beautiful!

pauldent

How did you know what factor to multiply with? Is there a standard procedure to find the factor?

kristoferkrus

how is it integrated without the sec?please respond i havent been taught about sec by my teachers so how can i integrate it withiut the sec(only with sinus, cosinus, tan and cot)?

orestis

But how do you come up with that idea?

ksk_futuremvrcreator

There wasn't actually any need to for the dx= du/sec(x)tan(x)+sec^2(x) arrangement. After stating u and getting du/dx, it's enough to make the suitable substitutions into the integral which is integral of (1/u)*du . However, thanks for simplifying the whole thing and making it easier to understand. I always mess these up when I come back to them now and then.

samchau

This can be done much easier by multiplying cos(x) on the numerator and denominator. Then numerator becomes d(sin(x)) and the denominator is 1-sin^2(x), then substitute u = sin(x)

guitarvijay