Integral of cos(ax)cos(bx) WITHOUT integration by parts

[@MuPrimeMath]

Has anyone counted the number of times that I said "right here"?

[@martinr3167]

import math print(math.floor(2*math.pi)) :v

[@VibingMath]

Yes! Thank you for reminding! We can use this method to handle sinsin, sincos and coscos type integration

[@alwaysuseless]

The math is great, and verbally the presentation is good. The one big improvement you could make in presentation is to AVOID erasing. Just move down and then eventually over to the right. The camera can follow you.

[@hudyakovnick]

Shouldn't "+C" be in the previous step? It is when we get rid of the integral, and then later in the end we simply calculate.

[@MuPrimeMath]

Yes, there should be a +C in the penultimate step. I opted not to write it until the final answer because it doesn't affect the calculations.

[@claireli88]

We can do the same way for integration of product of two sines of different angles too.

[@TheGiuse45]

I never use trigonometric formulas, I use complex numbers like a real OG

[@ShanBojack]

Can you elaborate

[@TheGiuse45]

@@ShanBojack instead of remembering all the trig formulas you can derive them by using the equivalence e^(ix)= cos(x) + i*sin(x) with x a real number. you can use this to make the cosine or sine, for example cos(x)= (e^(ix) + e^(-ix))/2. you can derive any trig formula from this

[@przemysawkwiatkowski2674]

How would you do it with integration by parts?

[@MuPrimeMath]

Choose one of the cosines to differentiate and one to integrate. After doing integration by parts twice, the resulting integral will be a constant multiple of the original. Then you can solve for the integral! This gives the result of b/(b^2-a^2)cos(ax)sin(bx) - a/(b^2-a^2)sin(ax)cos(bx) + C which is equivalent to the answer in the video.

[@viktor-kolyadenko]

a = b or a = -b (formally)?

[@MuPrimeMath]

Good point, those would be special cases!

[@diegogallego9370]

Thank you

[@ArifSolvesIt]

You could have presented a more complete solution by solving for the case a=b. You can still take the limit a goes to b in the final answer, but I think it is worth mentioning what happens when a=b.