Let be any function and its antiderivative. The set of all antiderivatives of is called **indefinite** **integral** of the function denoted by

Every two primitive functions differ by a constant . This means, if we know the one antiderivative of the function , then the another we can write in the form .

We do not have strictly rules for calculating the antiderivative (indefinite integral). The most antiderivatives we know is derived from the table of derivatives, which we read in the opposite direction.

**Table of basic integrals**

**Table of integrals of rational functions**

**Example 1**. Determine

**Solution**.

By the additivity and linearity property of the integral, we have

From the table of integral, we read

**Example 2**. Determine

**Solution**.

By the additivity property of the integral, we have

From the table of indefinite integrals, we read:

that is

**Example 3**. Determine

**Solution**.

Since , we can write

By the linearity and additivity property of the integral, we have

From the table of indefinite integrals we read:

that is

**Integration by substitution**

We will use the reverse chain rule of differentiation of composite functions and thus obtain the method which is called the method of substitution.

The chain rule:

The reverse of the chain rule:

**The integration rule**: if is a differentiable and continuous function on , then

where and .

Firstly, we need to choose a substitution to make. A substitution is not determined in advance, just by using the exercises we can discover the simplest way. Then we have to express the variable by and connect and . We change the integrand and with expressions by the variable and calculate the integral. The result we write as the function of the variable , that is, we return a substitution.

**Example 4**. Determine

**Solution**.

We choose a substitution It follows

and

Now, by changing with , we have:

From the linearity property of integrals we can write

From the table of integrals we read:

Finally, the result we need to write as the function of the variable :

that is

**Example 5**. Determine

**Solution**.

We choose a substitution . Therefore, we have

By replacing with and with we obtain:

From the table of integrals we read:

By returning a substitution, we obtain the finally result

**Example 6**. Determine

**Solution**.

We choose a substitution . It follows

By replacing with and with , we have

From the table of integrals we read:

that is

**Example 7**. Determine

**Solution**.

Since , we can write

We choose a substitution . It follows

Therefore, we have

From the table of integrals we read

By returning a substitution, we obtain

The same principle we use for calculating the definite integral. However, the result we do not need write as the function of the variable , because the result is a number. Instead, together with the integrand we are changing the lower and upper limit of integration.

**Example 8**. Determine

**Solution**.

We choose . It follows

The lower and upper limit are transformed in the following way:

Now we have:

From the relation , we can write , that is

By using the first fundamental theorem of calculus, finally we have:

**Integration by parts**

We cannot calculate all integrals by using the method of substitution. Therefore, for calculate the simple integral as we need to use different methods.

Integration by parts is the one useful method for calculating integrals. Let and be functions of the variable . The formula for derivative of the product

can be written in the equivalent form:

Thus we obtain:

or more simply

The formula above is called the** formula for integration by parts**.

**The method of the integration by parts**

- The integrand write as the product of functions and .
- Calculate the extra integral . The function must be chosen in the way that its integral is easy to determine.
- Calculate the derivative .
- Write the formula for integration by parts:

- Calculate the integral .

**Example 9**. Determine

**Solution**.

We choose and . Now we calculate the integral , that is

The derivative of is equal to

By using the formula for integration by parts we have

From the table of integrals we read: . Finally, we have

In some cases, integration by parts must be repeated.

**Example 10**. Determine

**Solution**.

We choose and . It follows

By using the formula for integration by parts, we have:

Now we choose and . It follows

that is

Finally, we have

that is

For the definite integral is valid the formula for integration by parts:

**Example 11**. Determine

**Solution**.

We choose and . It follows

By using the formula for integration by parts for definite integral and first fundamental theorem of calculus, we have: