Definite integrals: reverse power rule | AP Calculus AB | Khan Academy

Let's evaluate the definite integral from negative 3 to 5 of 4 dx. What is this going to be equal to? I encourage you to pause the video and try to figure it out on your own.

All right, so in order to evaluate this, we need to remember the fundamental theorem of calculus, which connects the notion of a definite integral and an antiderivative.

The fundamental theorem of calculus tells us that our definite integral from a to b of f of x dx is going to be equal to the antiderivative of our function f, which we denote with the capital F evaluated at the upper bound, minus our antiderivative evaluated at the lower bound.

So, we just have to do that right over here. This is going to be equal to... well, what is our antiderivative of 4? You might immediately say, well that's just going to be 4x. You could even think of it in terms of reverse power rule: 4 is the same thing as 4x to the 0. So, you increase 0 by 1, so it's going to be 4x to the first, and then you divide by that new exponent. 4x to the first divided by 1, well that's just going to be 4x.

So, the antiderivative is 4x. This is, you could say, our capital F of x. We're going to evaluate that at 5 and at negative three, and we're going to find the difference between these two.

What we have right over here, evaluating the antiderivative at our upper bound, that is going to be four times five. Then, from that, we're going to subtract evaluating our antiderivative at the lower bound, so that's four times negative three.

What is that going to be equal to? This is 20 and then minus negative 12. So, this is going to be plus 12, which is going to be equal to 32.

Let's do another example where we're going to do the reverse power rule. So, let's say that we want to find the definite integral going from negative 1 to 3 of 7x squared dx. What is this going to be equal to?

Well, what we want to do is evaluate what is the antiderivative of this, or you could say, if this is lowercase f of x, what is capital F of x? Well, the reverse power rule: we increase this exponent by 1. So, we're going to have 7 times x to the third, and then we divide by that increased exponent.

So, 7x to the third divided by 3, and we want to evaluate that at our upper bound and then subtract from that it evaluated at our lower bound. So, this is going to be equal to, evaluating it at our upper bound, it's going to be 7 times 3 to the third, I'll just write that 3 to the third over 3.

From that, we are going to subtract this capital F of x, the antiderivative evaluated at the lower bound, so that is going to be 7 times negative 1 to the third, all of that over 3.

So, this first expression, let's see, this is going to be 7 times 3 to the third over 3. This is 27 over 3, this is going to be the same thing as 7 times 9. So, this is going to be 63.

And this over here, negative 1 to the third power is negative 1, but then we're subtracting a negative, so this is just going to be adding. So this is just going to be plus 7 over 3. Plus 7 over 3, if we wanted to express this as a mixed number, seven over three is the same thing as two and one-third.

So when we add everything together, we are going to get 65 and one-third, and we are done.