# 7b. Difference Between Differentiation and Integration

## Integration Mini Video Lecture

## Video transcript

[Music...]

Let's now look at the difference between differentiation and integration.

Let's think of differentiation as going in the forward direction and integrate as going in the backwards direction.

Let's see how this works by **differentiating** 4 *x* to the power
of 7 and then **integrating** 4 *x* to the power of 7 and seeing how it is
different.

### Differentiating

So, in the first one, the `d/dx` of 4*x* to the 7th, just remembering my rules, I do
7 down to the front.

So it is going to be 7 times 4 is 28*x* to the
power 3 and I have a 7 here.

I have to subtract one for a differentiation so I am going to get 6.

So the derivative of 4*x* to the power 7 is 28*x* to
the power 6. `(d/dx 4x^7 = 28x^6)`

### Integration

Now, let's **integrate** 4 *x* to the power 7 and see the difference.

So I use the
integral sign, which is this extended "S", and I do 4 *x* to the power 7 and I do
*dx*. `(int 4x^7 dx)` This dx is important. Please don't forget it.

Now, I realize that this is the same as 4 integral x power 7 dx. `(4 int x^7 dx)`

All I'm doing is taking this 4 out the front so that you can see that I am just integrating the `x^7` by itself. Later on you will do this in your head.

Okay, so this is 4 and then times, now to integrate *x* to the 7th,
what I have to do is this. I have to add one to the index, so it's going to be *x* to the
power of 8. And then I have to divide by this new number *x* to the 8th over
8. And then I have to do plus *K*. `(4 int x^7 dx = 4(x^8)/8 + K)`

This *K* is a constant because if I differentiate
a constant I'll get zero. So I must always include this "plus *K*" when I'm doing an
indefinite integral.

Now, all I need to do is tidy it up. Four and the 8 cancel, so I'll
get a 2 in this position so I can do *x* to the power 8 over 2 and then plus *K*. `(4 int x^7 dx = (x^8)/2 + K)`

You have been watching The Difference Between Differentiation and Integration. A mini lecture brought to you by Interactive Mathematics.

[Music...]

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