ln|sinu|+c
The expression ln|sinu| + c represents the antiderivative of the function f(u) = |sinu|, where ln denotes the natural logarithm and c is a constant
The expression ln|sinu| + c represents the antiderivative of the function f(u) = |sinu|, where ln denotes the natural logarithm and c is a constant.
To find the antiderivative of f(u), we can break it into two cases: when sinu is positive and when sinu is negative.
Case 1: sinu > 0
In this case, |sinu| equals sinu. Therefore, we can write f(u) = sinu.
The antiderivative of sinu is -cosu. So, the antiderivative of f(u) in this case is -cosu + c.
Case 2: sinu < 0
In this case, |sinu| equals -sinu. Therefore, we can write f(u) = -sinu.
The antiderivative of -sinu is cosu. So, the antiderivative of f(u) in this case is cosu + c.
Combining both cases, the antiderivative of f(u) = |sinu| is given by:
ln|sinu| + c =
- cosu + c (when sinu > 0) or
cosu + c (when sinu < 0)
Note: The constant c represents the constant of integration and can be chosen arbitrarily.
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