How Many Antiderivatives Are There?

Calculus Level 1

True or false:

By double angle identities, we know that cos2x=2cos2x1=12sin2x \cos2x = 2\cos^2x - 1 = 1-2\sin^2 x.

2sin2xdx=cos2x+C=2cos2x+1+C=2cos2x+C \begin{aligned} \int 2\sin 2x \, dx &=& -\cos 2x + C \\ &=&- 2\cos^2x + 1 + C \\ &=& - 2\cos^2x + C \end{aligned}

2sin2xdx=cos2x+C=1+2sin2x+C=2sin2x+C \begin{aligned} \int 2\sin 2x \, dx &=& -\cos 2x + C \\ &=& -1 + 2\sin^2 x + C \\ &=& 2\sin ^2x + C \end{aligned}

From the two indefinite integrals below, we can see that 2cos2x+C=2sin2x+C  . - 2\cos^2x + C = 2\sin ^2x + C \; .

Cancelling off the arbitrary constant of integration CC, we obtain 2cos2x=2sin2x-2\cos^2 x = 2\sin^2x or equivalently cos2x=sin2x \cos^2 x =- \sin^2 x is true for all xx.

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