Integrate((1-Cos((Pi*x)/(2*a)))^2,x,0,a)

To solve the integral ∫((1-Cos((Pix)/(2a)))^2 dx from 0 to a, we can follow these steps:

1. First, expand the square in the integrand: (1 - Cos((Pix)/(2a)))^2 = 1 - 2Cos((Pix)/(2a)) + Cos^2((Pix)/(2a))

2. Now, we can split the integral into three separate integrals: ∫dx - 2∫Cos((Pix)/(2a)) dx + ∫Cos^2((Pix)/(2a)) dx from 0 to a.

3. The first integral is straightforward: ∫dx from 0 to a = [x]_0^a = a.

4. The second integral can be solved using the formula ∫Cos(u) du = Sin(u) + C. Here, u = (Pix)/(2a), so du = (Pi/2a) dx. Therefore, ∫Cos((Pix)/(2a)) dx = 2aSin((Pix)/(2a)) from 0 to a = 2a[0 - Sin(Pi/2)] = -2a.

5. The third integral, ∫Cos^2((Pix)/(2a)) dx from 0 to a, can be solved using the power-reduction identity: Cos^2(u) = 1/2 + 1/2Cos(2u). Therefore, ∫Cos^2((Pix)/(2a)) dx = 1/2∫dx + 1/2∫Cos(Pi*x/a) dx from 0 to a. The first integral is a/2, and the second integral is 0 (because the integral of Cos over a full period is 0).

6. Therefore, the original integral is a - 2a + a/2 = a/2.