trig integrals involving sine and cosine (calculus 2)

We will solve 8 integrals involving sine and cosine. Be sure to remember the trig identities in the video. Subscribe here for more calculus tutorials: bit.ly/just_calc
0:00 Integral of sin^2(x)*cos^5(x)
3:17 Integral of sin^3(x)/cos(x)
6:04 Integral of sin^2(3x)
8:16 Integral of cos(x)*cos(2x)
10:24 Integral of sin(2x)/sin(x)
11:12 Integral of sin(5x)*cos(2x)
13:18 Integral of (sin(x)+cos(x))^2
14:36 Integral of 1/(1-cos^2(x))
---------------------------------------------------------
This channel is dedicated to students who are taking AP calculus, GCSE, A-Level, year 12 maths, college calculus, or high school calculus. Feel free to leave calculus questions in the comment section and subscribe for future videos.
If you would like to support this channel, then check out my Patreon 👉 / blackpenredpen

[@HandscharGeorgeCostanza]

I like how calc isn't about learning calculus, it's about memorizing a bunch of identities that you should've learned years ago

[@carultch]

One trick is to learn to re-derive the identities from examining key points. As an example, I recognize from playing with my graphing calculator that both sin(x)^2 and cos(x)^2 can be re-written as a single trig function of twice the frequency. All that's left to do, is adjust the amplitude, phase, and vertical offset, to match it to the function of interest
To show how it works with sin(x)^2, we start by evaluating a few key points:
sin(0)^2 = 0
sin(pi/4)^2 = (sqrt(2)/2)^2 = 1/2
sin(pi/2)^2 = 1
sin(3*pi/4)^2 = 1/2
sin(pi)^2 = 0
This tells me that the function needs a midline of 1/2, a local maximum at (pi, 1), and a local minimum at (0, 0). Cosine starts on a peak and "coasts" downward, so we negate it to make it start on a valley and climb upwards. We then add 1/2 to it, and give it an amplitude of 1/2, to get our identity:
sin(x)^2 = -1/2*cos(2*x) + 1/2

[@Klaus-Schwab_Dictator]

😂

[@tzonic8655]

Even though It's been more than 2 years that ive finished my 3 calc Classes,i enjoy so much all your videos, thank tou so much for all ths help steve!

[@Vikingman2024]

Great help! Thank you!

[@rogerkearns8094]

Didn't know about that sinAcosB formula, thanks.

[@S21-rp5pe]

Thank you for the great explanation 🙏🏻 but I wanna ask about what should I do if I have the integration of sin * cos and they both have odd powers ,please help🙏🏻🙏🏻🙏🏻

[@kassidydavis5565]

I absolutely love how you explain and the examples that you put! You’re making me feel like I will be able to integrate one day 😂🤣❤️❤️🤞🤞🤞

[@buhle2419]

Thanks! I learnt a lot !!...

[@viliuskese3487]

you are a legend laughing at these integrals mate, good luck to you

[@tuinderdome]

Thank you very much sir!

[@mrmoonmini]

love the concept of sine integration ❤

[@Salamanca-joro]

Thank you man you are legned

[@lanomusambazi8654]

Thank you so much sir for the knowledge. I have learnt the technic of trig integrals. I'm kindly asking for trig integrals of CSC and COT. Secondly, how do we integrate when sec power is odd? I will appreciate your soonest response. Thank you.

[@nasaalfarah5652]

When secant power is odd, u=cosx. You take out a cos^2(x) from however much cosine you have. Finally you replace cos^2(x) with 1-sin^(x) and integrate and simplify at the end.

[@dioutoroo]

Thank you, Sir!

[@mrchiddy3628]

I have an next offer of calc Monday thick i will do it better for this tips

[@dfsgjlgsdklgjnmsidrg]

where do i learn all the identities

[@LiamFranklinFarang]

Trig 😂

[@Krageon-Offline]

where can i get that shirt

[@mimzim7141]

Or use the well known identity:
sin^2*cos^5= cos^5+cos^7

[@H3XED_OwO]

i hate trig but sometimes it's stupidity makes me cackle

[@mimzim7141]

Sorry i ment
sin^2*cos^5= cos^5-cos^7
of course.

[@jesemepardens9151]

8:18 Otherwise with the other formula ( cos(a)cos(b) = 1/2(cos(a-b)+cos(a+b)) ) this would give us :
1/2intg(cos(3x)+cos(-x))
= 1/2intg(cos(3x)-cos(x))
= 1/2(1/3sin(3x)-sin(x))
= 1/6sin(3x) - 1/2sin(x) +C (haha almost forgot that one)

[@hugodhia5707]

Is it actually the same with that in the video?

[@jesemepardens9151]

​@@hugodhia5707 I don't think so. Why would I have commented otherwise? The solution is the same though

[@hugodhia5707]

@@jesemepardens9151 I thought it is just a different way to get the same solution.
So, does that mean an integral can have more than one solution?

[@jesemepardens9151]

@@hugodhia5707 it is a different way. I don't know how but the result is the same ! The method I used is legit

[@tarfaibrahim912]

Identity memorising or understanding is the key points ....u can't jump to learn calculas without learning more basic trig algebra !!! But u can practice the ideas

[@seanliam5980]

too fast do step by step tutorial, tyia!

[@budaikax]

im so lost