单项式$x^n$的不定积分

It is often useful to be able to reverse differentiation, which means given a function $f$, we want to be able to find a function $g$ such that $g' = f$. We call this taking the anti-derivative. For now, if $f$ is a function of $x$, we'll write the anti-derivative as $Anti_x(f)$.

For example, suppose we know the acceleration $a(t)$ of an object as a function of time.

Okay, so now you know why taking anti-derivatives is useful. Let's go back to differentiation, then later thinking of how to reverse it.

Great!

Great!

Great!

So, we just found out $Anti_x((a+1)x^a) = x^{a+1}$.

We've just figured out the following rule!

ANTIDERIVATIVE OF POWERS OF X: $Anti_x(x^a) = \frac{x^{a+1}}{a+1}$

Perfect, so now we've found that $Anti_x(x^a) = \frac{x^{a+1}}{a+1}$, where $a$ is any constant.

...or have we?

In fact, the antiderivative of $x^{-1}$ is a special case, equal to $\ln x$ (the natural log of $x$.) It's okay if you're not yet very familiar with this case, however.

Now we can able to take antiderivatives of most polynomials: just take the antiderivative of each term, and add them all together!

At this point you should understand that the integral is the "reverse" operation of the derivative. This actually allows for some pretty powerful discoveries about integrating polynomials. For instance, it should be easy to see that

$\frac{d}{dx}(x^2 + 9x + 8) = 2x + 9$

However, the fact that the integral and derivative are reverse operations implies that $\int{\frac{d}{dx}f(x)dx} = f(x) + C$. Thus we can integrate both sides of the above equation to get

$\int{\frac{d}{dx}(x^2 + 9x + 8)dx} = \int{(2x + 9) dx} \Rightarrow$ $x^2 + 9x + 8 + C = \int{(2x + 9) dx}$

Differentiating this again gives the original equation. This means that $x^2 + 9x$ plus some real constant is the antiderivative of $2x + 9$.

This can be done more generically by using variables. We know that

$\frac{d}{dx}x^n = nx^{n - 1}$

This implies that, by integrating both sides of the equation,

$x^n + C = \int{nx^{n - 1}dx}$

Here is some practice for taking antiderivatives (integrals) of polynomials. GLHF!