# How Many Antiderivatives Are There?

Calculus Level 1

True or false:

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

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

\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 $- 2\cos^2x + C = 2\sin ^2x + C \; .$

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

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