Finding the Antiderivatives of a Function

Find all the antiderivatives of:

1. \(f(x) =0\)
2. \(g(\theta ) =-\sin \theta\)
3. \(h(x)=x^{1/2}\)

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 \(G(\theta ) =\cos \theta +C\).

330

(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.