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.