How Many Antiderivatives Are There?
True or false :
By double angle identities , we know that cos 2 x = 2 cos 2 x − 1 = 1 − 2 sin 2 x .
∫ 2 sin 2 x d x = = = − cos 2 x + C − 2 cos 2 x + 1 + C − 2 cos 2 x + C
∫ 2 sin 2 x d x = = = − cos 2 x + C − 1 + 2 sin 2 x + C 2 sin 2 x + C
From the two indefinite integrals below, we can see that − 2 cos 2 x + C = 2 sin 2 x + C .
Cancelling off the arbitrary constant of integration C , we obtain − 2 cos 2 x = 2 sin 2 x or equivalently cos 2 x = − sin 2 x is true for all x .
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2 solutions
And actually cos 2 x = 1 − sin 2 x . The difference between the two ' C 's accounts for the 1 .
More over after cancelling the constant term a square value cannot be equal to negative value.
Since the value of arbitrary constant C need not be the same for both the integrals, hence we can't cancel it and thus cos 2 x = − sin 2 x is not true for all x.