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# Antidifferentiation - Problem 2

###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

To find the antiderivative of a power of x, there is a rule you can follow: increase the power by 1, and divide by the new power. For example, the antiderivative of x is x^{2}/2 + c, and the antiderivative of x^{4} is x^{5}/5 + c. Remember to add the constant c at the end of the antiderivative.

More generally, if f'(x)=x^{n}, for some number n, its antiderivative is f(x)=x^{n+1}/(n+1).

Let’s try a harder problem. I want to find the anti-derivatives of first; little f(x) equals x to the 4th and then little f(x) equals x to the n any power function. Starting with x to the 4th. To find the anti-derivative of little f(x) equals x to the 4th. I need to find a function whose derivative is x to the 4th.

Remembering what derivatives do to power functions, they reduce the power by 1. So for example, the derivative with respect to x, of x to the 5th, I’ll get an x to the 4th out of this, because powers are reduced by 1. I’ll get 5x to the 4th. But that’s to exactly what I want. I want just x to the 4th. So what can I do to the original function to make the derivative come out just x to the 4th? I could divide it by 5. I can use 1/5x to the 5th. And my derivative will be 1/5 times 5x to the 4th. The 5 and the 1/5 cancel and I get x to the 4th.

So that tells me that one of the anti-derivatives for this function, is 1/5 x to the 5th. And I have a feeling that tells me once I have one anti-derivative of x to the 4th, I have all of them, just by adding a plus c. That gives me all the rest of them. Certainly, when c is 0, I get 1/5x to the 5th that is an anti-derivative. But c could be any other constant as well.

Let’s take a look at the second part; Anti-derivatives of f(x) equals x to the n. The next thing about this problem is once we solve it, we will have a formula for the anti-derivatives of any power function x to the n.

Well let’s use the same kind of ideas before. First, let’s recall we need to differentiate x to the n. What happens to the power? Comes out in front and goes down by 1. So what we want to end up with on the right side here, is x to then to n, not x to the n minus 1. So I have to start with one higher power. X to the n plus 1. Now when I differentiate this by the same rule, the n plus 1 comes out in front. X to the n, this goes down by 1. This is good, n plus 1 is just a constant and I have an x to the n like I need.

I can do a trick like I did before. I can make this constant come out 1 if I start with a function that has 1 over that constant in front of it. So 1 over n plus 1 times x to the n plus 1. Let’s see how that works. First of all this constant multiple is not touched by the derivative. We have 1 over n plus 1 times, and the derivative of x to the n plus 1 as we just saw, n plus 1 x to the n. These cancel and you get x to the n. Exactly what we need.

So this tells me that one anti-derivative of x to the n is one over n plus 1 x to the n plus 1. And of course, we can get all the rest of them by adding a constant. So this is a nice formula you can use any time you are anti-differentiating a power function. X to the n. The anti-derivatives of x to the n are 1 over n plus 1, x to the n plus 1, plus c.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

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