Basic Integration Examples
In this post I am going to go over some basic integration examples and integration rules. So let's say, if you want to integrate
for example, this will give you
where
c is constant. If you wish to integrate
this
will give you
or if
you want to integrate
that
will give you
so,
in all these examplesand
are the variable of integration. Make sure you
always add that particular variable based on variable of integration.
Now if you have a constant so let's say if we wish to integrate a constant
this is going to equal
so if
I want to integrate
this is going to be
if I wish to integrate let's say
that's going to be
Or
this
is going to be
Now another
rule that you need to be familiar with is the power rule.
let me give you some examples of using this power rule. So, let's say if you wish to find the antiderivative of
this is going to be
This
can also be written as
So, let's try some more examples using the power rule. Go ahead and find the antiderivative of
Go ahead and try to solve these problems.
So, for the first one
the
next one is
Now
what about the last one, what can we do for that example before you integrate it?
You need to rewrite it so we can rewrite it as
and
then use the power
and then we can rewrite the final answer as
so,
anytime you have a rational function like this one makes sure to rewrite it. So,
let's try another similar example. let's say if we want to integrate
try
that one. So first let's rewrite it as
and
then let's use the power rule so it's going to be
then this is equal to
that's
the final answer. Now you can also use the power rule when integrating radical
functions.
let's say if you want to find the antiderivative of
first, you need to rewrite it, this is
and
then use the power rule so this is going to be
so,
we can rewrite the final answer as
Now let's go over the antiderivative of six trigonometric functions.
Now
keep in mind if you know the derivative then you can know the antiderivative. The
derivative of sine is cosine, the derivative of cosine is negative sine so, the
derivative of negative cosine is positive sine.
Now
the antiderivative of
So, hopefully you had a chance to write it down.
Now
let's work on some example problems. Let’s try to integrate this function
so, this is going to be
so, the final answer is
and
if you know the formulas, it's very easy to integrate trigonometric functions that
look like this.
Here's another one that you could try to find the antiderivative
So, the
antiderivative of the given function is
so
this is the answer.
let's
work on one more example concerning trigonometric functions.
now
for this particular example, we need to rewrite it before we integrate it.
we still have the integration symbol and then the antider
ivative is
so
the final answer is going to be
it's gonna
be e to u divided by u prime plus C if u is a linear function or if u
prime is a constant, if u prime is not a constant this will not work. So let's work on some examples, let's say if we want to integrate
it's
going to be the same thing e to the 4π divided by the derivative of 4π which is
4 because this is a constant this will be the right answer
it's going to be the same thing
so
that's gonna be the answer. Now let's say if you want to integrate
it's
gonna be the same thing divided by its derivative of π which is 1 so it's simply
Now
let's see another example
this one won't work we can't say it's
so in this situation, this will not be the answer. If you try to find the derivative of this expression it will not give you

Now
let's move on to the antiderivative of logarithmicfunctions. The antiderivative of
if it
is a variable it would not work and u has to be a linear function.
so
let's say if we want to find the antiderivative of
it's going to be
let's say if we want to integrate
it's going to be
it's going to be
now
let's confirm the answer.
Let's find the derivative of this expression so if you were to differentiate
well, you will get the original answer the original problem. So in this case u is 3π-6 and ΓΊ
so in this case the 3 will cancel and this will
give us the original problemso it works.
Let's
try it one more example. What is the antiderivative of
what
I'm going to do is move the constant 5 to the front and then apply the same
process in the last problem
then this is going to be
the final answer be
that's it. Let's confirm it with differentiation. So let's find the derivative of this expression
so it's gonna be


and we can see that the 2 will cancel which give us the original problem
It’s very easy way and good explanation
ReplyDelete