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Antidifferentiation - Problem 3
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
For functions that are not power functions of x, finding the antiderivative is a matter of remembering specific derivatives. The derivative of the exponential function ex, for example, is itself, ex. So, the antiderivative of ex is ex + c.
The derivative of the natural log function ln(x) is 1/x, so the antiderivative of 1/x is ln|x| + c. The natural log must be taken of the absolute value of x because the function does not exist for positive values of x.
Let’s find two more anti-derivatives. I want to find the anti-derivatives of e to the x, and then of x to the -1. First anti-derivatives of little f(x) equals e to the x. Let’s recall the derivative of e to the x. It’s just e to the x. And let’s recall the anti-derivatives, what I’m looking for are functions that have e to the x as their derivative. Well clearly e to the x has itself as a derivative, so that’s going to be one anti-derivative.
But remember, that you can add a constant to this. And the derivative would still be e to the x. So this we can constitute the set of all the anti-derivatives of e to the x. So the anti-derivatives of little f(x) equals e to the x are capital F(x) equals e to the x plus c, very nice relationship. So every function of this form has e to the x as an anti-derivative.
Now this is a slightly harder problem; anti-derivatives of f(x) equals x to the -1. You might think but didn’t we already deal with this? We didn’t have a problem where we solved for the anti-derivatives of x to the n, and we got 1 over n plus 1, x to the n plus 1, plus c. The problem with this formula is it doesn’t work for n equals -1. So we are going to have to find some other way to anti differentiate this function. This formula doesn’t work, that’s the only case that it doesn’t work for.
Let’s recall that the derivative of natural log is 1 over x and that’s x to the -1. So we might have ourselves an anti-derivative. In fact we do. The only problem with lnx as an anti-derivative is its domain. Its domain is the set of positive numbers. So this formula only works for x larger than 0.
That’s a problem because x to the -1 has a domain of x greater than 0 or x less than 0. So what we would like to get is an anti-derivative that has the same domain. And that’s where this formula comes in. It can be shown that the derivative of the natural log of the absolute value of x is also 1 over x. Which is x to the -1. And this function, because you are taking the absolute value before you take the log. This function has domain, all real numbers except 0. Just like 1 over x. So this is the perfect anti-derivative to use. It is a much more general anti-derivative.
So based on that, I could start with this function. According to my formula its derivative is x to the -1. So this is an anti-derivative of x to the -1. And of course to get all the anti-derivatives, you just need to add a constant. So my anti-derivatives are natural log of the absolute value of x plus c. Don’t forget the absolute value. A lot of people forget this when they write the anti-derivatives. But the anti-derivatives of x to the -1 are natural log of the absolute value of x plus c.
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