what is the integral of absolute value of cosx
To find the integral of the absolute value of cos(x), you can break down the integral based on the intervals where cos(x) is positive and negative.
1. First, identify the intervals where cos(x) is positive. Since the absolute value of cos(x) is always positive or zero, we know that the integral of |cos(x)| over these intervals simplifies to just cos(x).
2. Next, identify the intervals where cos(x) is negative. In these intervals, the absolute value of cos(x) is equal to -cos(x). So, the integral of |cos(x)| over these intervals simplifies to -cos(x).
3. Combine the results obtained from steps 1 and 2 to get the complete integral.
For example, the interval for one complete cycle of cos(x) is [0, 2π]. Within this interval, cos(x) is positive in the interval [0, π] and negative in the interval [π, 2π].
To calculate the integral of |cos(x)| over [0, π]:
∫|cos(x)| dx = ∫cos(x) dx = sin(x) + C
To calculate the integral of |cos(x)| over [π, 2π]:
∫|cos(x)| dx = ∫-cos(x) dx = -sin(x) + C
Therefore, the integral of |cos(x)| is given by:
∫|cos(x)| dx = sin(x) - sin(x) + C = 2sin(x) + C, where C is the constant of integration.