So we’ve just looked at basic integration.
We’ve learned how integration is
different from differentiation.
We’ve also learned that integration helps
us find the area and their curves.
And we’ve learned the mathematical
method of adding one to the power
and dividing by the new power.
Like we did in differentiation
where then we moved on
to finding out how to
differentiate trig functions.
And so, we’re going to do exactly
the same with integration.
We’re going to start to look
at trigonometric functions,
and we’re going to look at some
different methods of integration
when you deal with more
Let me just talk you through
what we’re about to do.
We’re going to look at
So I’ll teach you the standard results
of lots of different trig functions.
We’re also going to do
integration by substitution,
which is a different type of method of
integration, makes life a little bit easier.
And then the last one is going
to be integration by parts,
which is fairly complicated functions,
and we’re going to talk
about this new technique,
which is splitting it
into parts to integrate.
The techniques are obviously going
to use a lot of trig functions
and the new techniques
that we’re about to learn.
And the application, you
know about this already.
We use integration to either
solve differential equations
or to find the area
and their curves.
So let’s just look at some standard
results that we’ve previously looked at.
We are now talking about differentiation.
So, when we differentiated,
if you had a function f of x,
and that was sine(x), you remember
that that differentiated to cos(x).
We also derived this result using
differentiation from first principles.
If you’re looking at a function cos(x),
that differentiates to -sine(x),
and we’ve done this
so many times.
Hopefully, you’ve just
managed to learn this.
Tan(x) differentiates to sec squared of x.
And again, we looked at the proofs for
this in the differentiation lectures.
Now, the next few, you don’t have to learn.
They’re just easy to
derive if you need to.
But just to give you the standard results,
if you’re ever looking for them in
your formula booklets or textbooks,
to -cosec squared of x.
Sec(x) goes to sec(x)tan(x).
Cosec of x differentiates
And these last two, we needed to learn.
So remember, we evaluated that e of x
differentiates to e of x, so it doesn’t change.
And ln of x differentiates
to 1 over x.
Now, if you think about what I
said in the previous lecture,
integration is just
So you could look at all the derivative
f' of x column here
and go backwards.
And if then you integrated cos of
x, so just look at the first line,
if you integrated cos of
x, you would get sine of x.
If you integrated -sine of
x, you would get cos of x.
So instead of going in this direction,
you’re going in the backward direction.
Let me just show that to you in
a table so it makes more sense.
So here’s what I’m trying to say.
I’m reversing the table
that we just looked at.
If you are asked to integrate cos
of x, that goes to sine of x.
If you’re asked to integrate sine
of x, this now goes to -cos of x.
Do you remember what cos
of x differentiated to?
-sine of x, and that’s where
that minus comes from.
It gets a little bit tricky so
we’re going to try and just
write something down for
ourselves so we can remember.
So once again, cos of x
integrates to sine of x.
And sine of x, this time,
integrates to -cos of x.
Sec squared of x
goes to tan of x,
just the way tan of x
differentiated to sec squared of x.
Sec squared of x
integrates to tan of x.
Nice and easy, e of x never changes.
So of e of x differentiated to e
of x; e of x integrates to e of x.
And remember what we said when you
differentiate ln of x, you get 1 over x.
So when we integrate 1 over x,
we’re going to go back to ln of x.
There’s quite a few results here
to familiarize yourselves with,
but we’ll keep this at the
front or on the side somewhere
so that we can see
it as we integrate.
Remember, this will all
come with practice.
So we’re going to do a lot of examples now
just to practice these new integrals.