逆微分

Let's say we have a function $f(x)$, and we want to know, is there any function $F(x)$ whose derivative is $f(x)$?

If there is, the function $F(x)$ is called an anti-derivative of $f(x)$.

Here are some interesting, non-obvious questions about anti-derivatives; think about them, and keep them in mind as you learn about anti-derivatives:

$\bullet$ Does every function have an anti-derivative?

$\bullet$ How many different anti-derivatives can a function have?

$\bullet$ Is there an easy way to find a function's anti-derivative?

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Let's find some anti-derivatives.

This example shows that a function can have more than one anti-derivative. Do you think a function can have more than two anti-derivatives? Think about it.

The concept of reversing differentiation is key to understanding antidifferentiation. This is because integration and differentiation are inverse operations. That is,

$f(x) = \frac{d}{dx}\int{f(x)dx}$

Antidifferentiating yields all possible functions which could be differentiated to get the original function. All of these possible functions differ by only a constant, labeled as $+C$ when antidifferentiation is complete. This is valid when back-differentiating because the derivative of any constant is $0$. In the form of an equation:

$\frac{d}{dx}(f(x)) = \frac{d}{dx}(f(x) + C)$

For instance, in the context of integrals, if $f(x)$ is one antiderivative of $f'(x)$:

$\int{f'(x)dx} = f(x)$

$\int{f'(x)dx} = f(x) + 1$

$\int{f'(x)dx} = f(x) + 2$

This continues on infinitely for any real constant. So any function that has one antiderivative has an infinite number of antiderivatives.

What is the opposite of a derivative?

The opposite of the derivative of a function is called the antiderivative. Given an expression, we want to figure out what that expression could be the derivative of. The process of solving for antiderivatives is called antidifferentiation or indefinite integration.

To find antiderivatives of basic functions, the following rules can be used $\int x^n\ dx = \frac{x^{n+1}}{n+1} + c$ as long as $n$ does not equal $-1$. This is basically the power rule for derivatives in reverse.

For example, if $f(x) = x^4$ , then an antiderivative of $f$ is $F(x) = \frac{x^5}{5 }$, which is basically found by reversing the power rule.

However, this is only one antiderivative of the function $f(x)=x^4$. Another antiderivative could be $F(x)= \frac{x^5}{5} + 3$. There are infinitely many functions whose derivatives equal a given function that may differ by only a constant. This is called the constant of integration ($C$). Thus, the antiderivative of $f(x)=x^4$ would be written in the form $F(x)=\frac{x^5}{5} + C$.

Okay, take the derivative of the listed functions. Now integrate the derivatives you found. Are they the same? If not, what's the difference? Just some food for thought.