Solving S_N Integral with Small Angle Formula

[ghostyc]

Homework Statement

$$S_N (x) = \frac{2}{\pi} \int_0^x \frac{\sin (2 N t )}{\sin (t)} \; d{t}$$

use suitable small angle formula to show$$S_N \Big( \frac{\pi}{2 N} \Big) = \frac{2}{\pi} \int_0^{\pi} \frac{\sin u}{u} d{u}$$

Homework Equations

i guess the suitable small angle formula is

$$\sin (\theta) \sim \theta$$

when $$(\theta)$$ is small...

The Attempt at a Solution

i have tried to do some substations but just can't get both numerator and denominator to the right thingany sugguestions will be appreciated

Thank YOU

 

[lurflurf]

change variable
u=2N*t
then
sin(small)~small

 

[ytys]

It looks like you already know they want you to assume $$\sin(t)=t$$. Your next step is to find an appropriate "u substitution." Try $$u = 2Nt$$ so that you have $$\int_0^{\pi/2N} \frac{\sin{(2Nt)}dt}{t} = \int_{0}^{?} \frac{\sin{u}}{u}du$$. Use algebra to find $$?$$ and $$du$$.

 

[ghostyc]

Hi there
I have tried this already (actually 7 days ago)
still using
$$u = 2 N t$$ i can get the correct limits but
i just can't justify that the bottom
$$t$$ just goes to $$u$$
how do i jusitfy that?

Thank you

++++++++++++++++++++++++++++
holly!

I got it
right after the click "post quick reply"...

THANK YOU ALL

:P

 

[wywong]

$$\frac{du}{u} = \frac{2Ndt}{2Nt} = \frac{dt}{t}$$