Integral of cos^2 x

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How to integrate cos^2 x using the addition formula for cos(2x) and a trigonometric identity.

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Why does cos 2x = cos^2 x - sin^2 x

Sometimes understanding where this stuff comes from can help more than throwing out a memorized formula.

It starts with the angle addition formula
cos(A+B) = (cos A cos B) - (sin A sin B)
A=x B=x substitute into angle addition formula
cos(x+x) = (cos x cos x) - (sin x sin x) collect terms and simplify
cos 2x = cos^2 x - sin^2 x
SEAN

postholedigger

Here is another way -- using integration by parts split this into the integral of /(cos x)(cos x)dx. Let cos x = u and cos x dx = dv. The du = -sin x dx and v = -sin x.
Then apply the formula /udv = uv - /vdu and we get = -sin x cos x -/(-sin x)(-sin x) dx = -sin x cos x -/(sin x)^2 dx = -sin x cos x -/(1 - (cos x)^2 ) dx
= -sin x cos x -/dx - / (cos x)^2 dx or 2 * /(cos x)^2 cx = -sin x cos x -/ dx +c or the final answer of - (1/2)(sin x cos x -x ) + c and since sin 2x = 2 sin x cos x
we get = - (( 1/4)( (sin 2x) - x)) + c. Relatively simple and straightforward.

thomasarch

My eyes are bleeding, too many substitutions

davideaccorto

Tbh, the intro music (Turkish March) made me instantly like this video. Good to see people with equally good taste in music and science. Looking forward to meeting you someday!

shishirmaharana

That was really helpful, thanks a lot

tj_

You could’ve just used the power reduction formula from the very beginning and factored out a 1/2 then all you have to integrate is 1/2 ∫1+cos(2x)dx Which is really easy to integrate.

davebacknolaliki

I can see clearly now the mist is gone. Thank you sir

ZolileZicwele

finally got what i was looking for man thanks

rishabhlakhara

Thankyou thankyou so so so much sir 😊😊😊🦚🙏🏻🙏🏻 Hare krishna 🦚❣️❣️you cleared my huge doudt 🙇🙇🙇

SAHITY_A

This comes to recommendation video from Youtube, thank you Youtube for knowing that I've done with every channel on my following list...

Maverick

Let it go bro. ... U make easy things complicated.... U aren't for teaching ❤️

prakharpratapsingh

Lmaooo I was trying to use the reduction method on this

hypnoticmoai

Very easy question...this is a very basic question in Indian education and is taught in 11th class basics

superduper

And why did u use the reverse chain rule ?

manishvelagala

cos^2 x = 1/2(1 + cos2x) and then integrate simply

azmatsahi

You're doing the lord's work, my man

georgianfishbowl

2:24 compensate for 2x, reverse chain rule, can you elaborate and explain?
Thank You.

s-r

shouldn't there be an easier way to do this?

angelinebena

Dude your voice is too low.. i didn't get the point that how you've taken 1/2 from cos2x/s.. can you and anyone explain me that... PLEASE I REQUEST🙏🙏

monusonu-cm

this is so cool
thank you dude
integration is pain

rishabhlakhara

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