B Why Do Different Methods Yield Different Results for ∫0 T sin²(ωt) dt?

[Const@ntine]

Hi! I came across a proof in my physics textbook (amperage=wattage/area), and it contained this integration: ∫₀ ^T sin²(ωt) dt

The whole thing: 1/T∫₀ ^T sin²(ωt) dt = 1/T(t/2 + sin2ωt/2ω)|^T ₀ = 1/2

I didn't remember how to integrate that, so I went back to check my notes, and look at it at Wolfram or some other sites. But the problem is, I get different results. Same for the other sites I checked.

Wolfram: http://www.wolframalpha.com/input/?i=integrate+sin^2(ω*t)

Cymath (it's a simpler form, but the theory is the same): https://www.cymath.com/answer?q=int(sin(x)^2,x)

So I went to try my hand at it:

∫₀ ^T sin²(ωt) dt

u = ωt <=> du = ωdt
sin²(x) = 1/2 - cos(2x)/2

1/ω∫₀ ^T 1/2 - cos2u/2 du = 1/ω [ ∫₀ ^T1/2 du - ∫₀ ^T cos2u/2 du]

We break that into two integrals:

1/ω∫₀ ^T1/2 du = 1/ω (u/2)|^T ₀ = 1/ω (ωt/2)|^T ₀ = T/2

1/ω∫₀ ^T cos2u/2 du

k = 2u <=> dk = 2du

1/ω∫₀ ^T cosk/4 dk = 1/4ω∫₀ ^T cosk dk = 1/4ω(sink)|^T ₀ = 1/4ω(sin2u)|^T ₀ = 1/4ω(sin2ωt)|^T ₀ = sin(2ωT)/4ω

But, ω = 2π/T so sin(2ωT)/4ω = 0

In the end, we have: 1/T*(T/2 - 0) = 1/2

The end result is the same, but I wonder which version is correct:

Book: 1/T∫₀ ^T sin²(ωt) dt = 1/T(t/2 + sin2ωt/2ω)|^T ₀

Other: 1/T∫₀ ^T sin²(ωt) dt = 1/T(t/2 - sin2ωt/4ω)|^T ₀

Any help is appreciated!

PS: I know that I also have to change the T & 0 when changing the integrating factor (t, then u, then k), but it's been a few years since I've done that, and have forgotten how exactly, so I justkept them the same and reverted the u & k back to their original t-related forms when the time came to find the end result.

 

[mjc123]

"Other" is correct, as you have shown by your integration. There is a factor of 1/2 that comes from sin²x = 1/2(1-cos2x), and another that comes from ∫-cos2xdx = (sin2x)/2. The book seems to have forgotten one of these.

 

[Const@ntine]

"Other" is correct, as you have shown by your integration. There is a factor of 1/2 that comes from sin²x = 1/2(1-cos2x), and another that comes from ∫-cos2xdx = (sin2x)/2. The book seems to have forgotten one of these.

Thanks for the confirmation!