The key to understanding antiderivatives is
to understand derivatives. Every formula for a
derivative, f′(x)=g(x), can be read both ways. The function g
is the derivative of f, but f is also an antiderivative of
g. In the following video, we use this idea to generate
antiderivatives of many common functions. Learn these antidifferentiation rules.
One really nice thing about antiderivatives
is this: once you antidifferentiate, you can always
check your answer by differentiating -- it's like having the answer
right there (as long as you know your differentiation rules).
DO: Is the antiderivative of
f(x)=cos(5x) the function F(x)=15sin(5x)?
Check to see.
Table of common antiderivatives -- notice that you already know these because you know how to
differentiate.
FunctionAntiderivativeCommentsxnxn+1n+1+CAs long as n≠−1exex+C Your favorite
antiderivative, too!1xln(|x|)+CDon't forget the
absolute value!cos(x)sin(x)+CBe careful with the
sign.sin(x)−cos(x)+CBe careful with the
sign.sec2(x)tan(x)+Csec(x)tan(x)sec(x)+Ccsc2(x)−cot(x)+Ccsc(x)cot(x)−csc(x)+C11+x2tan−1(x)+C1√1−x2sin−1(x)+C