What is the integral of sin6x ?

To find the integral of sin^6(x), you can use the power reduction formula for sine raised to an even power. The formula is:

sin^n(x) = (1/2^n) * C(n, k) * (-1)^k * cos(2kx) * sin^(n-2k)(x),

where:

n is the power to which sin is raised (in this case, n = 6),
k is a positive integer less than or equal to n/2,
C(n, k) is the binomial coefficient "n choose k," which is equal to (n!)/(k!(n-k)!), and
The formula is applied for all valid values of k such that n - 2k is greater than or equal to 0.
In this case, n = 6, so we will use the formula for n = 6:

sin^6(x) = (1/2^6) * C(6, 0) * (-1)^0 * cos(20x) * sin^6(x) + (1/2^6) * C(6, 1) * (-1)^1 * cos(21x) * sin^4(x) + (1/2^6) * C(6, 2) * (-1)^2 * cos(22x) * sin^2(x).

Now, let's calculate the values:

C(6, 0) = 1
C(6, 1) = 6
C(6, 2) = 15

So, the integral becomes:

(1/2^6) * 1 * cos(0) * sin^6(x) + (1/2^6) * 6 * (-1) * cos(2x) * sin^4(x) + (1/2^6) * 15 * 1 * cos(4x) * sin^2(x)

Simplify each term:

(1/64) * sin^6(x) - (3/32) * cos(2x) * sin^4(x) + (15/64) * cos(4x) * sin^2(x)

So, the integral of sin^6(x) is:

(1/64) * sin^6(x) - (3/32) * cos(2x) * sin^4(x) + (15/64) * cos(4x) * sin^2(x) + C,

where C is the constant of integration.