Find all the antiderivatives of:

Solution (a) Since the derivative of a constant function is $0,$ all the antiderivatives of $f$ are of the form $F(x) =C,$ where $C$ is a constant.

(b) Since $\dfrac{d}{d\theta}\cos \theta = -\sin \theta$, all the antiderivatives of $g(\theta) = -\sin \theta$ are of the form $G(\theta ) =\cos \theta +C$.

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(c) The derivative of $\dfrac{2}{3}x^{3/2}$ is $\left( \dfrac{2}{3}\right) \ \left( \dfrac{3}{2}x^{\hbox{$\frac{3}{2}-1$}}\right) =x^{1/2}$.

So, all the antiderivatives of $h(x)=x^{1/2}$ are of the form $H(x) ={{\dfrac{2}{3}}x^{3/2}+C}$, where ${C}$ is a constant.