Basic Integration Examples

The process of finding the integral is integration. Many students thinks that the integration is difficult. I also think the same when I was studying. If you are familiar with the differentiation than solving the integration is easy, because integration is the reverse process of derivative. That's why it is also called antiderivative. Hopefully, this post is helpful for you if you want to learn some basic rule of integration and 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 examplesandare 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 

Now let's discuss about integrating exponential functions. For example

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 4𝒙 divided by the derivative of 4𝒙 which is 4 because this is a constant this will be the right answer

now let's say if we want to integrate 

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

 now try this one, what's the antiderivative of

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 3𝒙-6 and ΓΊ is 3. So this is going to be 

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

so in this case once again we need to apply in this formula
in this situation
and the derivative of u is
and we can see that the 2 will cancel which give us the original problem

so that's how you can confirm it.
















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