Definite integrals: reverse power rule

let's evaluate the definite integral from negative 3 to 5 of 4 DX what is this going to be equal to and I encourage you to pause the video and try to figure it out on your own alright 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 so 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 - our antiderivative evaluated at the lower bound so we just have to do that right over here so this is going to be equal to well what is the antiderivative of 4 well 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 4 X 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 exponents 4 X 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 and we're going to evaluate that at 5 and at negative 3 and we're going to find the difference between these two so what we have right over here evaluating the antiderivative at our upper bound that is going to be 4 times 5 and then from that we're going to subtract evaluating our antiderivative at the lower bound so that's 4 times negative 3 4 times negative 3 and what is that going to be equal to so 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 indefinite or 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 what is if this is lower case 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 so evaluating it at our upper bound it's going to be 7 times 3 to the 3rd I'll just write that 3 to the 3rd over 3 and then from that we are going to subtract this capital f of X the antiderivative evaluated the lower bound so that is going to be 7 times negative 1 to the 3rd all of that over 3 and so this first expression let's see this is going to be 7 times 3 to the 3rd 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 3rd power is negative 1 but then we're subtracting a negative so this is going to be adding and 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 7 over 3 is the same thing as 2 and 1/3 so when we add everything together we are going to get 65 and one-third and we are done