Given the string (In), n>=1, I2n=integral sin^(2n) xdx, x=0 to x=pi/2, calculate I2.

In other words, we'll have to calculate the 2nd term of
the string, using the formula of the general term of the
string.

To determine the 2nd term, we'll have to evaluate
the definite integral of the function (sin x)^(2n).

We'll
put I2n = I2

I2 = Int (sin x)^2
dx

We'll apply the formula for the half
angle:

(sin x)^2 = (1 - cos
2x)/2

I2 = Int (1 - cos
2x)dx/2

I2 = Int dx/2 - (1/2)*Int cos 2x
dx

I2 = x/2 - (1/2)*[(sin
2x)/2]

I2 = x/2 - (sin
2x)/4

We'll apply Leibniz-Newton formula for evaluating the
definite integral I2.

I2 = F(pi/2) -
F(0)

I2 = pi/4 - (sin pi)/4 - 0/2 + (sin
0)/2

I2 =
pi/4

The 2nd term of the string is I2 =
pi/4.