How to reconcile the fact that the antiderivative of sin(x) cos(x) has two possible answers?So...

Quintin Stafford

Quintin Stafford

Answered

2022-07-01

How to reconcile the fact that the antiderivative of sin(x) cos(x) has two possible answers?
So here is the problem.
sin ( x ) cos ( x ) d x = 1 2 sin 2 ( x ) + c 1 = 1 2 cos 2 ( x ) + c 2
This fact doesn't make much sense to me. How do you reconcile the fact that there are two possibilities (notice that these answers aren't only different by a constant)?
What are some other functions that have more than one antiderivatives (not only different by a constant)?

Answer & Explanation

Donavan Scott

Donavan Scott

Expert

2022-07-02Added 22 answers

Explanation:
You wrote:
(notice that these answers aren't only different by a constant)
But they do differ by a constant. If you subtract one solution from the other, you have
( 1 2 sin 2 ( x ) + c 1 ) ( 1 2 cos 2 ( x ) + c 2 )
which is equal to 1 2 ( sin 2 ( x ) + cos 2 ( x ) ) + c 1 c 2 and using the trig identity sin 2 ( x ) + cos 2 ( x ) = 1, all of this is equal to just 1 2 + c 1 c 2 which is a constant.
Oakey1w

Oakey1w

Expert

2022-07-03Added 2 answers

Explanation:
Since   sin 2 ( x ) + cos 2 ( x ) = 1, if we let c 1 = c 2 1 2 so we'd have
1 2 sin 2 ( x ) + c 1 = 1 2 sin 2 ( x ) + ( c 2 1 2 ) = 1 2 sin 2 ( x ) + ( c 2 1 2 ) = 1 2 ( 1 cos 2 ( x ) ) + ( c 2 1 2 ) = c 2 1 2 cos 2 ( x )

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