Integration is a reverse process of differentiation, hence we can also call it as inverse differentiation. It is a technique of finding a function with its derivative. In this article, we have jotted down some basic integration formulas. We will also solve some examples using these formulas.

## Integration Formulas

Some basic integration formulas are given below:

1.

2.

3.

4.

5. , where

6.

7.

8.

9.

10.

11. , where , and

12.

Now, we will see how to use the above formulas to find the integrals.

## Example 1

Find the integral of the following function:

### Solution

Since the above function contains power or exponent, hence we will use the following formula to compute the integral:

, where

In this question, n = 6. Now, we will put this value in the above formula:

Simplifying the above equation will give us the following answer:

## Example 2

Find the integral of the following function:

### Solution

Since the above function contains variables with exponents and constant both, hence we will use the following formulas to compute the integral:

For exponent: , where

For constant:

But before applying these formulas, we will rewrite the question using integration sum rule like this:

We will separately integrate the three terms like this:

Now, we will combine these answers again to write the integral of the entire function:

## Example 3

Find the integral of the following function:

### Solution

Using the integration sum rule, we can rewrite the above function like this:

We will treat both the terms as separate functions for now. We know that and .

Combining the integrals of both the terms will give us the following answer:

## Example 4

Find the integral of the following function:

### Solution

First, we will convert this radical function into exponential form like this:

Now, we will use substitution method to solve the problem. Suppose (x - 1) = u. This means that and du= dx. Substituting these values in the above equation will give us the following expression:

Now, we will use the formula , where to find the integral:

In the end, we will substitute u = x - 1, again in the above expression to get the final answer:

## Example 4

Find the integral of the following function:

### Solution

First, we will rewrite the above example by using sum rule of integration like this:

It means that we will integrate the three terms separately. Let us first integrate the first term. To do so, we will convert this radical function into exponential form like this:

Now, we will use substitution method to find the integral of this term. Suppose 2x = u. This means that and . We can easily find the value of dx which is equal to . Substituting these values in the above equation will give us the following expression:

Move the fraction before the integral sign like this:

Now, we will use the formula , where to find the integral:

In the end, we will substitute u = 2x again in the above expression to get the final integral of this term:

Now, we will integrate the remaining two terms. We will use the formula , where again to find the integral of the second term .

The third term is a constant, so we will use the formula to integrate it.

In the end, we will combine the integrals of all the three terms to write the final answer as shown below:

## Example 5

Find the integral of the following function:

### Solution

The question in this example is an exponential function. Hence, we will use the formula , where , and to integrate it.

=

=

## Example 6

Find the integral of the following function:

### Solution

In this problem, we will use the substitution method to find the integral. Suppose 5x - 2 = u. If u = 5x - 2, then . From this, we can calculate du and dx.

Now, we will substitute these values in the above function like this:

Move the fraction before the integral sign like this:

Now, we will use the formula . Substituting these values in the above function will give us the final answer:

Substitute u = 5x - 2 again in the above expression to get the following answer:

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