integrate
Cos^7(2x)
To integrate cos^7(2x), you can use the power reduction formula to simplify the expression. The power reduction formula states that cos^2(x) = (1 + cos(2x))/2.
Let's apply the power reduction formula multiple times to cos^7(2x):
cos^7(2x) = (cos^2(2x)) * (cos^2(2x)) * (cos^2(2x)) * cos(2x)
Using the power reduction formula for cos^2(2x), we have:
cos^7(2x) = [(1 + cos(4x))/2] * [(1 + cos(4x))/2] * [(1 + cos(4x))/2] * cos(2x)
Expanding this expression, we get:
cos^7(2x) = [(1 + 3cos(4x) + 3cos^2(4x) + cos^3(4x))/8] * cos(2x)
Now, let's substitute cos^2(4x) with the power reduction formula for cos^2(2x) again:
cos^7(2x) = [(1 + 3cos(4x) + 3(1 + cos(8x))/2 + cos^3(4x))/8] * cos(2x)
Simplifying further:
cos^7(2x) = [(1 + 3cos(4x) + 3/2 + 3/2cos(8x) + cos^3(4x))/8] * cos(2x)
Now we have an expression that can be integrated term by term.
The integral of cos(2x) is sin(2x), and integrating the other terms can be done using the power rule for integration.
By integrating each term, we can determine the integral of cos^7(2x).