Antiderivative of | sin ⁡ ( x ) | Because ∫ 0 π sin ⁡...

Janet Forbes

Janet Forbes

Answered

2022-07-02

Antiderivative of | sin ( x ) |
Because 0 π sin ( x ) d x = 2 ,, then 0 16 π | sin ( x ) | d x = 32..
And Wolfram Alpha agrees to this, but when I ask for the indefinite integral | sin ( x ) | d x ,,
Wolfram gives me cos ( x ) s g n ( sin ( x ) ) + c ..
However, [ cos ( x ) s g n ( sin ( x ) ) ] 0 16 π = 0 ,,
So what's going on here? What is the antiderivative of | sin ( x ) | ?

Answer & Explanation

Mateo Carson

Mateo Carson

Expert

2022-07-03Added 15 answers

Step 1
Let f ( x ) = 0 x | sin x | d x = ( 1 cos x ) , 0 x π ..
since the integrand is π- periodic, we can extend the formula for f ( x ) = f ( π ) + f ( x π ) , π x 2 π
Step 2
And so on. you can verify that f ( n π ) = 2 n  for all integer  n ..
in particular f ( 16 π ) = 32..
Savanah Boone

Savanah Boone

Expert

2022-07-04Added 5 answers

Step 1
For x [ n π , ( n + 1 ) π ], we get
(1) 0 n π | sin ( t ) | d t = 2 n
and n π x | sin ( t ) | d t = 0 x n π sin ( t ) d t (2) = 1 cos ( x n π )
Step 2
Piecing (1) and (2) together yields (3) 0 x | sin ( t ) | d t = 1 cos ( x π x / π n ) + 2 x / π n

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