Original subtitles

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Today we're going to be talking about how to evaluate an indefinite integral.

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And in this particular problem we've been given the integral of 3 x squared plus 2 x plus one dx and

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this is the most basic type of indefinite integral that we can deal with only because we'll only really

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need to reverse the power rule process that we learn from derivatives in order to evaluate it.

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You remember that when we were dealing with derivatives and we were using the power rule to find derivatives.

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What we have is something like this or we can even take this example here.

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If we had 3 x squared and we wanted to take its derivative.

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Remember that power rule told us that we would bring the exponent down here in front and we'd multiply

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it by whatever coefficient was already there.

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So in other words we'd get three times two.

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We'd leave the X there and then then we'd subtract 1 from the exponent.

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We'd end up with two minus one.

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And of course the result of that was a derivative of 6 x.

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When we're dealing with an indefinite integral that's just a polynomial like this.

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We're reversing the power rule process so this time instead of kind of dealing with the coefficient

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first and then the exponent second like we did with the derivative.

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This time we're going to be dealing with the exponent first then the coefficient and what I mean by

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that is instead of subtracting 1 from the exponent we're going to be adding 1 to the exponent.

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So the X is going to stay here.

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And instead of two minus one we're going to say two plus one.

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Then we take our new exponent in this case.

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Two plus 1 and we divide our existing coefficient by this new exponent.

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So our existing coefficient is three we leave that there but we divide by the new exponent which is

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2 plus 1.

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So this is going to be our integral and the result of this of course here we'll get 3 divided by three.

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So that's just going to be one.

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And this will disappear here.

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And all we're left with is now x to the third two plus one is three.

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We're left with X to the third as the integral of 3 x squared and that's how we're going to deal with

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each of these polynomial terms.

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So let's take a look at this integral here.

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We already know that the integral of 3 x squared is x cubed.

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And remember we're just taking this term by term one term at a time when we have a polynomial.

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We only need to deal with one term at a time.

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Now we move on to 2 x and again here for 2x what we're going to do is leave the existing coefficient

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to we've got the X here.

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This is X to the first power.

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So we're going to add 1 to the exponent we're gonna get one plus one and then we're going to divide

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our existing coefficient by the new exponent and put one plus one there.

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Now this is the technical slow way of doing it.

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As you practice you'll get much faster at doing this you won't have to add 1 to the exponent like this.

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Write it out and then divide by this expanded exponent year you'll get faster at it.

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But I just wanted to show you guys how it worked.

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Now when you get to the integral of a constant like this we have one here you can just know that the

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integral of one is X in a similar way.

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The integral of 3 is 3 x so the integral of 3 would be 3 x the integral of 1 would be 1 x or just x.

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If you had the integral of pi which is also a constant the integral would be pi x you get the idea.

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But the way to remember this it's the same reversal of power rule.

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This isn't one.

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It's one times x to the 0.

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Remember that X to the 0 is equal to 1 because anything raised to the power of 0 is 1.

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So this is really 1 times x to the zero power.

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And essentially what we're just doing here is the same process we've done here before we say plus the

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existing coefficient 1.

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We leave the X here like this.

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We add one to the exponent so we get 0 plus 1 and we divide by our new exponent.

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So we say 0 plus 1 here and you'll see how that's going to simplify to just X like we saw before.

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So now we simplify this we get x cubed plus two divided by two is 1.

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So that's going to disappear.

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We get X squared plus x squared.

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And then here zero plus 1 is 1 1 divided by one is one.

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So this will disappear also.

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And we're left with X to the first power so we just have plus X like this.

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And remember we saw that here when we took the integral of one we got 1 x or just X and you can see

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we ended up with X here so that's technically why the integral of a constant is just that constant multiplied

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by x.

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So this is our indefinite integral except that our last step is to add c whenever you're dealing with

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an indefinite integral.

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You always have to add c which we call the constant of integration.

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The reason that we have to add that is because there are several possible answers for the integral of

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this particular function.

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The reason is because of this constant value here or any constant value remember that when you take

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the derivative of a constant the derivative is zero.

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Right.

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If we take the derivative of one we get zero if we take the derivative of four we get zero.

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If we take the derivative of Pi we get zero.

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The derivative of any constant is zero.

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So when we're reversing the process we don't know if the original function had some extra constant added

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onto it.

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Our original function could have been x cubed plus x squared plus X plus 4.

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And if it was when we took its derivative we would have ended up with this value here 3 x squared plus

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2 x plus 1.

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But there would have been no sign of the constant the constant would have just disappeared.

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So this polynomial here has to include this plus C value because what it tells us is that the integral

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of this original function here is this polynomial Plus it could have had some constant attached to it.

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Any of these constants for maybe just 0.

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But either way we have to account for the possibility that there was a constant here so whenever we

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have this indefinite integral we always have to add the constant of integration.

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Once we do we can say that this is our final answer.