How Do You Solve the Integral of (sin 37x / sin x) from 0 to π?

[jd12345]

Homework Statement

Find the value of ∫ ( sin 37x / sin x )dx - from 0 to π

Homework Equations

The Attempt at a Solution

Well i tried doing it by parts but it makes its more complicated. I tried to expand sin 37x as sin(36+1)x but could not solve it. I am lost

 

[Ray Vickson]

Homework Statement

Find the value of ∫ ( sin 37x / sin x )dx - from 0 to π

Homework Equations

The Attempt at a Solution

Well i tried doing it by parts but it makes its more complicated. I tried to expand sin 37x as sin(36+1)x but could not solve it. I am lost

$$\sin(nx) = \frac{1}{2i} ( X^n - Y^n), \text{ where } X = e^{ix}, \: Y = 1/X.$$
Thus, $\sin(nx) = \sin(x) [X^{n-1} + X^{n-2}Y + \cdots + Y^{n-1}],$ so
$$\sin(nx)/\sin(x) = X^{n-1} + X^{n-2}Y + \cdots + Y^{n-1}.$$

RGV