Basic Antiderivatives Exercises

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Here we’ll practice basic antiderivative rules.

Which of the following are antiderivatives of $\sin (x)\cos (x)$ with respect to $x$ ? Check all that apply.

$\frac {1}{2}\sin ^2(x)+2$ $\frac {1}{2}\sin ^2(x)\cos ^2(x)$ $\sin (2x)-9$ $\frac {-1}{2}\cos ^2(x)+3$

Which of the following are antiderivatives of $0$ with respect to $x$ ? Check all that apply.

$x$ $3$ $0$ $x^0$

Which of the following are antiderivatives of $\frac {2x}{(1-x^2)^2}$ with respect to $x$ ? Check all that apply.

$\frac {x}{1-x}+3$ $\frac {1}{1-x^2}+2$ $\frac {1}{(1-x^2)^2}+2$ $\frac {1}{1-x^2}+3$

Does this violate our result about antiderivatives being constant shifts of each other? If not, why not?